esaim: m2an...esaim: m2an 48 (2014) 1807–1857 esaim: mathematical modelling and numerical analysis...

ESAIM: M2AN 48 (2014) 1807–1857 ESAIM: Mathematical Modelling and Numerical AnalysisDOI: 10.1051/m2an/2014021 www.esaim-m2an.org

ON SOME IMPLICIT AND SEMI-IMPLICIT STAGGERED SCHEMESFOR THE SHALLOW WATER AND EULER EQUATIONS

R. Herbin1, W. Kheriji

2and J.-C. Latche

2

Abstract. In this paper, we propose implicit and semi-implicit in time finite volume schemes forthe barotropic Euler equations (hence, as a particular case, for the shallow water equations) and forthe full Euler equations, based on staggered discretizations. For structured meshes, we use the MACfinite volume scheme, and, for general mixed quadrangular/hexahedral and simplicial meshes, we usethe discrete unknowns of the Rannacher−Turek or Crouzeix−Raviart finite elements. We first showthat a solution to each of these schemes satisfies a discrete kinetic energy equation. In the barotropiccase, a solution also satisfies a discrete elastic potential balance; integrating these equations over thedomain readily yields discrete counterparts of the stability estimates which are known for the continuousproblem. In the case of the full Euler equations, the scheme relies on the discretization of the internalenergy balance equation, which offers two main advantages: first, we avoid the space discretization of thetotal energy, which involves cell-centered and face-centered variables; second, we obtain an algorithmwhich boils down to a usual pressure correction scheme in the incompressible limit. Consistency (in aweak sense) with the original total energy conservative equation is obtained thanks to corrective termsin the internal energy balance, designed to compensate numerical dissipation terms appearing in thediscrete kinetic energy inequality. It is then shown in the 1D case, that, supposing the convergence of asequence of solutions, the limit is an entropy weak solution of the continuous problem in the barotropiccase, and a weak solution in the full Euler case. Finally, we present numerical results which confirmthis theory.

Mathematics Subject Classification. 35Q31, 65N12, 76M10, 76M12.

Received January 1st, 2013. Revised January 16, 2014.Published online October 10, 2014.

1. Introduction

The objective pursued in this work is to develop and analyze a class of efficient numerical schemes for thesimulation of compressible flows at all Mach number regimes. To this purpose, our basic choice is to extendalgorithms that are classical in the incompressible framework, namely pressure correction schemes based on(inf-sup stable) staggered discretizations.

Keywords and phrases. Finite volumes, finite elements, staggered, pressure correction, Euler equations, shallow-water equations,compressible flows, analysis.

1 Aix-Marseille Universite, CNRS, Centrale Marseille, I2M, UMR 7373, 13453 Marseille, France. [email protected] Institut de Radioprotection et de Surete Nucleaire (IRSN), BP 3, 13115 Saint-Paul-lez-Durance cedex, [email protected]; [email protected]

Article published by EDP Sciences c© EDP Sciences, SMAI 2014

http://dx.doi.org/10.1051/m2an/2014021

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1808 R. HERBIN ET AL.

The fractional step strategy involving an elliptic pressure correction step has been recognized to yield algo-rithms which are not limited by stringent stability conditions (such as CFL conditions based on the celerity ofthe fastest waves) since the first attempts to build “all flow velocity” schemes in the late sixties [23] or in theearly seventies [24]; these algorithms may be seen as an extension to the compressible case of the celebrated MACscheme, introduced some years before [25]. These seminal papers have been the starting point for the develop-ment of numerous schemes, using staggered finite volume space discretizations [4, 6, 34, 35, 38, 41, 47, 64–69,71],colocated finite volumes [2,10,32,33,36,37,39,43,48–51,54,57,59,61,70] or finite elements [3,46,52,72]. Algorithmsproposed in these works may be essentially implicit-in-time, and the pressure correction step is then an ingredi-ent of a SIMPLE-like iterative procedure, or only semi-implicit, with a single (or a limited number of) predictionand correction step(s), as in projection methods for incompressible flows (see [7, 60] for seminal works and [19]for a review of most of the variants). The schemes which we propose in the present paper fall in this latter class.

We first deal with the barotropic Euler equations, which read:

∂t ρ+ div(ρu) = 0, (1.1a)∂t (ρu) + div(ρu ⊗ u) + ∇p = 0, (1.1b)ρ ≥ 0, p = ℘(ρ) = ργ , (1.1c)

where t stands for the time, ρ, u, p, are the density, velocity, pressure in the flow, and γ > 1 is a coefficientspecific to the considered fluid. For p = ργ , γ = 2 and ρ = h, we also get the shallow water (or Saint-Venant)equations.

Then, we address the full Euler equations, which are obtained from the complete Navier–Stokes equations

∂tρ+ div(ρu) = 0, (1.2a)

∂t(ρu) + div(ρu ⊗ u) + ∇p− div(τ (u)) = 0, (1.2b)

∂t(ρE) + div(ρE u) + div(pu) = 0, (1.2c)

p = (γ − 1) ρ e, E =12|u|2 + e, (1.2d)

by neglecting the viscous stress tensor, i.e. setting τ (u) = 0. In this system, E and e stand for the total energyand the internal energy in the flow, respectively.

Both problems (1.1) and (1.2) are supposed to be posed overΩ×(0, T ), whereΩ is an open bounded connectedsubset of Rd, 1 ≤ d ≤ 3, and (0, T ) is a finite time interval. System (1.1) (resp. (1.2)) is complemented by initialconditions for ρ and u (resp. ρ, e and u); these initial conditions are denoted by ρ0 and u0 (resp. ρ0, e0 andu0), with ρ0 > 0 and e0 > 0. In both cases, we consider the boundary condition u · n = 0 at any time and a.e.on ∂Ω, where n stands for the normal vector to the boundary.

This paper is organized as follows. We begin by describing the space discretizations (Sect. 2). We then addressthe barotropic case in Section 3; we introduce a fully implicit scheme and a pressure correction scheme (Sects. 3.1and 3.2 resp.). We prove the stability of each algorithm and their consistency in the one-dimensional case, in theLax–Wendroff sense. We finally propose in Section 4 a pressure correction scheme for the full Euler equations. Wefirst give the general form of the algorithm (Sect. 4.2). The scheme solves the internal energy balance, which offersthree advantages: first, we avoid the space discretization of the total energy, which involves cell-centered andface-centered variables; second, we obtain an algorithm which boils down to a usual pressure correction scheme inthe incompressible limit; third, this relation implies that the internal energy remains positive. Consistency (againin the Lax–Wendroff sense) with the original total energy conservative equation is obtained thanks to correctiveterms in the internal energy balance, designed to compensate numerical dissipation terms appearing in thediscrete kinetic energy inequality. It is then shown in the 1D case (Sect. 4.3), that, supposing the convergence ofa sequence of solutions, the limit is a weak solution of the continuous problem. Finally, we present some numericaltests in Section 4.4 for the case of the Euler equations (note that the numerical study of the correction pressure

SEMI-IMPLICIT STAGGERED SCHEMES FOR THE EULER EQUATIONS 1809

scheme in the barotropic case was performed in [27]). In several theoretical developments, we are lead to use aderived form of a discrete finite volume convection operator (for instance, typically, a convection operator forthe kinetic energy, possibly with residual terms, obtained from the finite volume discretization of the convectionof the velocity components); the proofs of various related discrete identities are given in the Appendix.

2. Meshes and unknowns

Let M be a decomposition of the domain Ω, supposed to be regular in the usual sense of the finite elementliterature (see e.g. [8]). The cells may be:

– for a general domain Ω, either convex quadrilaterals (d = 2) or hexahedra (d = 3) or simplices, both types ofcells being possibly combined in a same mesh,

– for a domain whose boundaries are hyperplanes normal to a coordinate axis, rectangles (d = 2) or rectangularparallelepipeds (d = 3) (the faces of which, of course, are then also necessarily normal to a coordinate axis).

By E and E(K) we denote the set of all (d−1)-faces σ of the mesh and of the element K ∈ M respectively. Theset of faces included in Ω (resp. in the boundary ∂Ω) is denoted by Eint (resp. Eext); a face σ ∈ Eint separatingthe cells K and L is denoted by σ = K|L. The outward normal vector to a face σ of K is denoted by nK,σ.For K ∈ M and σ ∈ E , we denote by |K| the measure of K and by |σ| the (d − 1)-measure of the face σ. For1 ≤ i ≤ d, we denote by E(i) ⊂ E and E(i)

ext ⊂ Eext the subset of the faces of E and Eext respectively which areperpendicular to the ith unit vector of the canonical basis of Rd.

The space discretization is staggered. In the case of rectangular or orthogonal parallelepipedic meshes, weuse the Marker-And Cell (MAC) scheme [24, 25]. For mixed simplicial or quadrilateral/hexahedral meshes, weuse the discrete unknowns of the Crouzeix−Raviart [9] and Ranacher–Turek [58] finite element spaces; however,the associated finite element formulation is not used here (but it would readily provide a discretization of thediffusion terms, in the Navier–Stokes case [1, 16]).

For all these space discretizations, the degrees of freedom for the scalar unknowns are associated with thecells of the mesh M; these are the discrete pressure and density unknowns, and, for the full Euler equations,the internal energy unknowns, which are denoted by:{

pK , ρK , eK , K ∈ M}.

Let us then turn to the degrees of freedom for the velocity (i.e. the discrete velocity unknowns).

– Rannacher−Turek or Crouzeix−Raviart discretizations – The discrete velocity unknowns are located atthe center of the faces of the mesh, and represent the average of the velocity through the face. The set ofdiscrete velocity unknowns reads:

{uσ,i, σ ∈ E , 1 ≤ i ≤ d}.

– MAC discretization – The degrees of freedom for the ith component of the velocity are located at the centerof the faces σ ∈ E(i), so that the set of discrete velocity unknowns reads:{

uσ,i, σ ∈ E(i), 1 ≤ i ≤ d}.

We now introduce a dual mesh, which will be used for the finite volume approximation of the time derivativeand convection terms in the momentum balance equation.

– Rannacher−Turek or Crouzeix−Raviart discretizations – For the RT or CR discretizations, the dualmesh is the same for all the velocity components. When K ∈ M is a simplex, a rectangle or a cuboid, forσ ∈ E(K), we define DK,σ as the cone with basis σ and with vertex the mass center of K (see Fig. 1). Wethus obtain a partition of K in m sub-volumes, where m is the number of faces of the mesh, each sub-volume


Dσ

Dσ′

σ′ = K|MK

L

M

|σ|σ=

K|L

ε = Dσ |D

σ ′ Dσ

K

L

σ = K|L

σ′

ε = Dσ|Dσ′

Figure 1. Notations for control volumes and dual cells – Left: Finite Elements (the presentsketch illustrates the possibility, implemented in our software (CALIF3S [5]), of mixing simpli-cial (Crouzeix−Raviart) and quadrangular (Rannacher−Turek) cells) – Right: MAC discretiza-tion, dual cell for the y-component of the velocity.

having the same measure |DK,σ| = |K|/m. We extend this definition to general quadrangles and hexahedra,by supposing that we have built a partition still of equal-volume sub-cells, and with the same connectivities;note that this is of course always possible, but that such a volume DK,σ may be no longer a cone; indeed, ifK is far from a parallelogram, it may not be possible to build a cone having σ as basis, the opposite vertexlying in K and a volume equal to |K|/m. The volume DK,σ is referred to as the half-diamond cell associatedwith K and σ.For σ ∈ Eint, σ = K|L, we now define the diamond cell Dσ associated with σ by Dσ = DK,σ ∪DL,σ; for anexternal face σ ∈ Eext ∩ E(K), Dσ is just the same volume as DK,σ.

– MAC discretization – For the MAC scheme, the dual mesh depends on the component of the velocity. Forthe ith component, the dual cells are associated to the faces perpendicular to the ith unit vector of thecanonical basis of Rd, i.e. to the faces of E(i) (which is, of course, consistent with the location of the velocitydiscrete unknowns). A MAC dual cell only differs from the corresponding RT or CR one by the choice of thehalf-diamond cell, which, for K ∈ M and σ ∈ E(K), is now the rectangle or rectangular parallelepiped ofbasis σ and of measure |DK,σ| = |K|/2.

We denote by |Dσ| the measure of the dual cell Dσ, by ε = Dσ|Dσ′ the face separating two diamond cells Dσ

and Dσ′ , and by E(Dσ) the set of the faces of Dσ.Finally, we need to deal with the impermeability (i.e. u · n = 0) boundary condition. Since the velocity

unknowns lie on the boundary (and not inside the cells), these conditions are taken into account in the definitionof the discrete spaces. To avoid technicalities in the expression of the schemes, we suppose throughout this paperthat the boundary is a.e. normal to a coordinate axis, (even in the case of the RT or CR discretizations), whichallows to simply set to zero the corresponding velocity unknowns:

for i = 1, . . . , d, ∀σ ∈ E(i)ext, uσ,i = 0. (2.1)


Therefore, there are no discrete velocity unknowns on the boundary for the MAC scheme, and there are onlyd − 1 discrete velocity unknowns on each boundary face for the CR and RT discretizations, which depend onthe orientation of the face. In order to be able to write a unique expression of the discrete equations for bothMAC and CR/RT schemes, we introduce the set of faces E(i)

S associated with the degrees of freedom of eachcomponent of the velocity (S stands for “scheme”):

E(i)S =

∣∣∣∣∣∣E(i) \ E(i)

ext for the MAC scheme,

E \ E(i)ext for the CR or RT schemes.

Similarly, we unify the notation for the set of dual faces for both schemes by defining:

E(i)S =

∣∣∣∣∣∣E(i) \ E(i)

ext for the MAC scheme,

E \ E(i)ext for the CR or RT schemes,

where the symbol ˜ refers to the dual mesh; for instance, E(i) is thus the set of faces of the dual mesh associatedwith the ith component of the velocity, and E(i)

ext stands for the subset of these dual faces included in theboundary. Note that, for the MAC scheme, the faces of E(i) are perpendicular to a unit vector of the canonicalbasis of Rd, but not necessarily to the ith one.

Note that general domains can easily be addressed (of course, with the CR or RT discretizations) by redefining,through linear combinations, the degrees of freedom at the external faces, so as to introduce the normal velocityas a new degree of freedom.

3. Implicit and semi-implicit schemes for the barotropic equations

We study two schemes for the numerical solution of System (1.1) which differ by the time discretization: thefirst one is implicit, and the second one is a non-iterative pressure-correction scheme introduced in [14]. Thislatter algorithm (and, by an easy extension, also the first one) was shown in [14] to have at least one solution,to provide solutions satisfying ρ > 0 (and therefore also p > 0) and to be unconditionally stable, in the sensethat its (their) solution(s) satisfies an inequality corresponding to the control in L∞(0, T ) of the integral of thediscrete entropy over the domain. In this section, we complement this work in several directions. For the implicitscheme, we obtain the following results.

– First we pass from a (discrete) global (i.e. integrated over Ω) entropy balance to (discrete) local balanceequations. Precisely speaking, a discrete kinetic energy balance is established on dual cells, while a discretepotential elastic balance is established on primal cells.These equations yield the stability of the scheme (i.e. the same global entropy conservation as in [14]) by asimple integration in space (i.e. summation over the primal and dual control volumes).

– Second, in one space dimension, the limit of any convergent sequence of solutions to the scheme is shown tobe a weak solution to the continuous problem, and thus to satisfy the Rankine–Hugoniot conditions.

– Finally, passing to the limit in the discrete kinetic energy and elastic potential balances, such a limit is alsoshown to satisfy the usual entropy inequality.

For the pressure correction scheme, the results are essentially the same: the scheme is unconditionally stable,and the passage to the limit in the scheme shows that if a sequence of approximate solutions obtained with thescheme is assumed to converge to some limit, then the predicted and end-of-step velocities necessarily tend tothe same function, and that the limit is a weak solution to the problem, satisfying the entropy inequality. Thenumerical study of this scheme (which is the only one implemented in practice) is performed in [27]. It confirmsthe present theoretical study: in particular, the scheme is observed to converge to weak entropy solutions ofRiemann problems, with an approximately first order rate; in addition, it yields qualitatively correct solutionsfor CFL numbers much larger than one.


3.1. An implicit scheme

3.1.1. The scheme

Let us consider a uniform partition 0 = t0 < t1 < . . . < tN = T of the time interval (0, T ), and letδt = tn+1 − tn for n = 0, 1, . . . , N − 1 be the constant time step. We consider an implicit-in-time scheme, whichreads in its fully discrete form, for 0 ≤ n ≤ N − 1:

∀K ∈ M,|K|δt

(ρn+1

K − ρnK

)+∑

σ∈E(K)

Fn+1K,σ = 0, (3.1a)

For 1 ≤ i ≤ d, ∀σ ∈ E(i)S ,

|Dσ|δt

(ρn+1

Dσun+1

σ,i − ρnDσun

σ,i

)+∑

ε∈E(Dσ)

Fn+1σ,ε un+1

ε,i − |Dσ|(ΔM u

)n+1

σ,i+ |Dσ| (∇p)n+1

σ,i = 0, (3.1b)

∀K ∈ M, pn+1K = ℘

(ρn+1

K

)=(ρn+1

K

)γ, (3.1c)

where the terms introduced for each discrete equation are defined hereafter.Equation (3.1a) is obtained by discretizing the mass balance (1.1a) over the primal mesh, and Fn+1

K,σ standsfor the mass flux across σ outward K, which, because of the impermeability condition, vanishes on externalfaces and is given on the internal faces by:

∀σ = K|L ∈ Eint, Fn+1K,σ = |σ| ρn+1

σ un+1K,σ ,

where un+1K,σ is an approximation of the normal velocity to the face σ outwardK. This latter quantity is defined by:

un+1K,σ =

∣∣∣∣∣un+1σ,i e(i) · nK,σ for σ ∈ E(i) in the MAC case,

un+1σ · nK,σ in the CR and RT cases,

(3.2)

where e(i) denotes the ith vector of the orthonormal basis of Rd. The density at the face σ = K|L is approximatedby the upwind technique:

ρn+1σ =

∣∣∣∣∣ ρn+1K if un+1

K,σ ≥ 0,

ρn+1L otherwise.

(3.3)

Note that the positivity of the density in (1.1c) is not enforced in the scheme but results from the aboveupwind choice (see e.g. [16], Lem. 2.1).

We now turn to the discrete momentum balance (3.1b). In order to obtain the desired estimates on theapproximate solution such as a discrete kinetic energy inequality, the discretization of the momentum balancemust be performed in a way that is compatible with the discretization of the mass balance equation. Hence wemust carefully choose the values ρn+1

Dσand ρn

Dσas functions of the primal unknowns (ρn+1

K )K∈M and (ρnK)K∈M

and the fluxes on the dual faces Fn+1σ,ε as functions of the fluxes on the primal faces Fn+1

K,σ . The values ρn+1Dσ

andρn

Dσ, which approximate the density on the face σ at time tn+1 and tn respectively, are given by the following

weighted average:

for σ = K|L ∈ Eint, for k = n and k = n+ 1, |Dσ| ρkDσ

= |DK,σ| ρkK + |DL,σ| ρk

L. (3.4)

Let us then detail the discretization of the convection term. The first task is to define the discrete mass fluxthrough the dual face ε outward Dσ, denoted by Fn+1

σ,ε ; the guideline for its construction is that a finite volumediscretization of the mass balance equation over the diamond cells, of the form

∀σ ∈ E , |Dσ|δt

(ρn+1

Dσ− ρn

Dσ

)+∑

ε∈E(Dσ)

Fn+1σ,ε = 0 (3.5)


must hold in order to be able to derive a discrete kinetic energy balance (see Sect. 3.1.1). For the MAC scheme,the flux on a dual face which is located on two primal faces is the mean value of the sum of the fluxes on thesetwo primal faces, and the flux of a dual face located between two primal faces is again the mean value of the sumof the fluxes on these two primal faces [28]. In the case of the CR and RT schemes, for a dual face ε included inthe primal cell K, this flux is computed as a linear combination (with constant coefficients, i.e. independent ofthe cell) of the mass fluxes through the faces of K, i.e. the quantities (Fn+1

K,σ )σ∈E(K) appearing in the discretemass balance (3.1a). We refer to [1, 15] for a detailed construction of this approximation. Let us remark thata dual face lying on the boundary is then also a primal face, and the flux across that face is zero. Therefore,the values un+1

ε,i are only needed at the internal dual faces; we choose them to be centered (in fact, the upwindchoice is also covered by our analysis, see the comments on the numerical diffusion below); so, for ε = Dσ|Dσ′ ,un+1

ε,i reads:

un+1ε,i =

un+1σ,i + un+1

σ′,i

2· (3.6)

The quantity (ΔMu)n+1σ,i stands for a possible stabilizing diffusion term which may be written under a finite

volume form over any diamond cell Dσ associated with the ith component of the velocity:

−|Dσ| (ΔMu)n+1σ,i =

∑ε=Dσ |Dσ′

ν hd−2ε

(un+1

σ,i − un+1σ′,i

), (3.7)

where hε is a characteristic dimension of the face ε, and ν stands for a non-negative coefficient, possibly dependingon a power of hε. Note that external faces are excluded in the sum at the right-hand side, which means that thepossible associated diffusion flux is set to zero. Note also that (ΔMu)n+1

σ,i is usually (i.e. for general meshes)not consistent with a Laplace operator. Moreover, the upwind scheme

un+1ε,i =

∣∣∣∣∣ un+1σ,i if Fn+1

σ,ε ≥ 0,

un+1σ′,i otherwise,

can also be written under this form, since in this case, the convection term may be written as:

(Fn+1

σ,ε un+1ε,i

)(up)= Fn+1

σ,ε

un+1σ,i + un+1

σ′,i

2+

12

∣∣Fn+1σ,ε

∣∣ (un+1σ,i − un+1

σ′,i

)for ε = Dσ|Dσ′ . (3.8)

Hence the upwind choice is included in the formulation (3.1b), (3.6), with a numerical diffusion term definedby (3.7), setting

ν hd−2ε =

12

∣∣Fn+1σ,ε

∣∣ , (3.9)

and ν behaves as hε in this case (provided that the density and the velocity are uniformly bounded).The introduction of a numerical diffusion of the form (3.7) presents two advantages:

– On one hand, if we assume that the coefficient ν is such that C1hαε ≤ ν ≤ C2h

αε with C1, C2 ∈ R+ and

0 < α < 2, we obtain a weak L2(H1) control of the velocity which is sufficient, at least in one space dimension,to pass to the limit in the scheme (see Sect. 3.1.3). This is not the the case for a pure (i.e. without additionaldiffusion term) upwind discretization, because ν vanishes with the mass flux (i.e. the normal velocity).Note however that the situation is different if we now assume that the approximate velocity u satisfies a BVestimate: the convergence analysis of Section 3.1.3 then still holds in the centered or upwind case, withoutrequiring any additional diffusion.

– On the other hand, this formalism may prepare for a stabilization strategy which could be less diffusive thanthe upwind choice, choosing for instance ν on the basis of an a posteriori analysis of the local regularity ofthe solution [20, 21, 40].


The last term (∇p)n+1σ,i stands for the ith component of the discrete pressure gradient at the face σ. The gradient

operator is built as the transpose of the natural discrete divergence operator which is defined by

|K| (divu)K =∑

σ∈E(K)

|σ| uK,σ. (3.10)

In the CR and RT case, the duality between the divergence and gradient operators simply reads:∑K∈M

|K| pK (divu)K +∑σ∈E

|Dσ| uσ · (∇p)σ = 0.

This duality relation may be rewritten so as to fit both the CR/RT scheme and the MAC scheme as follows:

∑K∈M

|K| pK(divu)K +d∑

i=1

∑σ∈E(i)

S

|Dσ| uσ,i (∇p)σ,i = 0. (3.11)

Therefore, on any internal face, the components of the gradients are given by:

for σ = K|L ∈ Eint, (∇p)n+1σ,i =

|σ||Dσ|

(pn+1L − pn+1

K ) nK,σ · e(i).

Note that, because of the impermeability boundary conditions, the discrete gradient is not defined at the externalfaces.

Finally, the initial approximations for ρ and u are given by the average of the initial conditions ρ0 and u0

on the primal and dual cells respectively:

∀K ∈ M, ρ0K =

1|K|

∫K

ρ0(x) dx,

for 1 ≤ i ≤ d, ∀σ ∈ E(i)S , u0

σ,i =1

|Dσ|

∫Dσ

(u0(x))i dx.

(3.12)

3.1.2. Estimates

We begin with an estimate on the velocity which is a discrete equivalent of the kinetic energy balance. Recallthat in the continuous setting, the kinetic energy balance is formally obtained by multiplying the ith componentof the momentum balance equation (1.1b) by the ith component ui of u; this yields for 1 ≤ i ≤ d, using themass balance equation (1.1a) twice:

∂t

(12ρu2

i

)+ div

((12ρu2

i

)u

)+ (∂xip) ui = 0,

and thus, summing over the components:

∂t(ρEk) + div(ρEk u

)+ ∇p · u = 0, with Ek =

12|u|2. (3.13)

In the discrete setting, this multiplication must be performed on the dual mesh, since the velocity unknownsare defined on the faces. This is the reason why we chose the fluxes on the faces of the dual mesh in such a waythat a discrete mass balance equation holds on the dual grid cells, thus allowing us to use Lemma A.2 (whichperforms the discrete equivalent of the above formal computations) on the dual mesh.


Lemma 3.1 (Discrete kinetic energy balance, implicit scheme). A solution to the system (3.1) satisfies thefollowing equality, for 1 ≤ i ≤ d, σ ∈ E(i)

S and 0 ≤ n ≤ N − 1:

12|Dσ|δt

[ρn+1

Dσ(un+1

σ,i )2 − ρnDσ

(unσ,i)

2]+

12

∑ε=Dσ |Dσ′

Fn+1σ,ε un+1

σ,i un+1σ′,i + |Dσ| (∇p)n+1

σ,i un+1σ,i = −Rn+1

σ,i , (3.14)

where

Rn+1σ,i =

|Dσ|2 δt

ρnDσ

(un+1

σ,i − unσ,i

)2+

⎡⎣ ∑ε=Dσ|Dσ′

ν hd−2ε

(un+1


)⎤⎦ un+1σ,i . (3.15)

Proof. Let us multiply equation (3.1b) by the corresponding velocity unknown un+1σ,i ; this yields

T convσ,i + TΔ

σ,i + T∇σ,i = 0

with:

T convσ,i =

⎡⎣ |Dσ|δt

(ρn+1

Dσun+1

σ,i − ρnDσun

σ,i

)+

∑ε=Dσ |Dσ′

Fn+1σ,ε un+1

ε,i

⎤⎦ un+1σ,i ,

TΔσ,i =

⎡⎣ ∑ε=Dσ |Dσ′

ν hd−2ε (un+1

σ,i − un+1σ′,i )

⎤⎦ un+1σ,i ,

T∇σ,i = |Dσ| (∇p)n+1

σ,i un+1σ,i .

Applying Lemma A.2 with P = Dσ, we get from the identity (A.9) that

T convσ,i =

12|Dσ|δt

[ρn+1

Dσ(un+1

σ,i )2 − ρnDσ

(unσ,i)

2]

+12

∑ε=Dσ |Dσ′

Fn+1σ,ε un+1

σ,i un+1σ′,i +

|Dσ|2 δt

ρnDσ

(un+1

σ,i − unσ,i

)2.

We then note that

TΔσ,i +

|Dσ|2 δt

ρnDσ

(un+1

σ,i − unσ,i

)2 = Rn+1σ,i ,

where Rn+1σ,i is defined by (3.15), which concludes the proof of (3.14). �

Let us now define the elastic potential P :

P(z) =∫ z

0

℘(s)s2

ds i.e. P(z) =

⎧⎪⎨⎪⎩zγ−1

γ − 1if γ > 1,

ln(z) if γ = 1,(3.16)

and let H be the function defined over (0,+∞) by

H(z) = z P(z) =

⎧⎨⎩zγ

γ − 1if γ > 1,

z ln(z) if γ = 1.(3.17)

It may easily be checked that zH′(z)−H(z) = ℘(z); therefore, by a formal computation detailed in the appendix(see (A.3)), multiplying (1.1a) by H′(ρ) yields:

∂t

(H(ρ)

)+ div

(H(ρ)u

)+ p div(u) = 0. (3.18)

We now derive a discrete analogue of this relation.


Lemma 3.2 (Discrete potential balance). Let H be defined by (3.17). A solution to the system (3.1) satisfiesthe following equality, for K ∈ M and 0 ≤ n ≤ N − 1:

|K|δt

[H(ρn+1

K ) −H(ρnK)]

+∑

σ∈E(K)

|σ| H(ρn+1σ ) un+1

K,σ + |K| pn+1K (divun+1)K = −Rn+1

K , (3.19)

with:

Rn+1K =

|K|2 δt

H′′(ρn+ 12

K ) (ρn+1K − ρn

K)2 +12

∑σ=K|L

|σ| (un+1K,σ )− H′′(ρn+1

σ ) (ρn+1σ − ρn+1

K )2, (3.20)

where ρn+ 1

2K ∈ [min(ρn+1

K , ρnK),max(ρn+1

K , ρnK)], ρn+1

σ ∈ [min(ρn+1σ , ρn+1

K ),max(ρn+1σ , ρn+1

K )] for all σ ∈ E(K),and, for a ∈ R, a− ≥ 0 is defined by a− = −min(a, 0). Note that, since the function H is convex, Rn+1

K isnon-negative.

Proof. Let us multiply the discrete mass balance (3.1a) by H′(ρn+1K ). The result is then a consequence of

Lemma A.1 with P = K, using the fact that zH′(z)−H(z) = ℘(z) and that ρn+1σ is the upwind choice between

ρK and ρL in the remainder term RK,δt. �

Summing (3.13) and (3.18) yields:∂tη + div

((η + p)u

)= 0,

where η = ρEk +H(ρ) is the entropy of the system. This relation is only valid for regular solutions, and shouldbe replaced by an inequality to take into account the presence of shocks (see relations (3.34)–(3.35)). Integratingover Ω and using the boundary conditions yields:

ddt

∫Ω

η(x, t) dx ≤ 0 (for regular solutions,ddt

∫Ω

η(x, t) dx = 0),

and, for t ∈ (0, T ), ∫Ω

η(x, t) dx ≤∫

Ω

η(x, 0) dx.

The following proposition states a discrete analogue to this relation.

Proposition 3.3 (Global discrete entropy inequality, existence of a solution). Assume that the initial densityρ0 is positive. Then, there exists a solution (un, ρn) 0≤n≤N to the scheme (3.1), and, for 1 ≤ n ≤ N , ρn > 0and the following inequality holds:

12

d∑i=1

∑σ∈E(i)

S

|Dσ| ρnDσ

(unσ,i)

2 +∑

K∈M|K| H(ρn

K) + Rn ≤ C0, (3.21)

where C0 ∈ R+ only depends on the initial conditions, and Rn is the following non-negative remainder whichdepends on the space and time translates of the unknowns:

Rn =d∑

i=1

n∑k=1

⎡⎢⎣12

∑σ∈E(i)

S

|Dσ| ρk−1Dσ

(ukσ,i − uk−1

σ,i )2 + δt∑

ε=Dσ|Dσ′∈E(i)S

ν hd−2σ (uk

σ,i − ukσ′,i)

2

⎤⎥⎦+γ

2

n∑k=1

δt∑

σ=K|L∈Eint

|σ| (ρkσ,γ)γ−2 |uk

K,σ| (ρkK − ρk

L)2,

(3.22)

with ρkσ,γ equal to either ρk

K or ρkL and such that (ρk

σ,γ)γ−2 = min((ρk

K)γ−2, (ρkL)γ−2

).


Remark 3.4. For γ > 1, the function H is positive and increasing over (0,+∞). The inequality (3.21) thusreadily provides an estimate on the unknowns.

This is still true also for γ = 1, since in this case H(z) = z ln z and therefore H(z) ≥ −1/e, ∀z ∈ (0,+∞), andH is increasing over (1/e,+∞). In fact, in order to get the usual formulation of an estimate, we may rephrase theinequality (3.21) by changing the expression of H to H(z) = max(z log(z), 0) and adding |Ω|/e to the constantC at the right-hand side.

Proof. Let us give the proof of Proposition 3.3. The positivity of the density is a consequence of the propertiesof the upwind choice (3.3) for ρ [16], Lemma 2.1; note that it may also be proved applying Lemma A.1 withψ(s) = 1

2 (s−)2 and P = K.

Let us then sum equation (3.14) over the components i and the faces σ ∈ E(i)S , equation (3.19) over K ∈ M,

and, finally, the two obtained relations. Since the discrete gradient and divergence operators are dual withrespect to the L2-inner product (see (3.11)), noting that the conservative fluxes vanish in the summation, weget, for 1 ≤ k ≤ N :

12

d∑i=1

∑σ∈E(i)

S

|Dσ|δt

[ρk

Dσ

(uk

σ,i

)2 − ρk−1Dσ

(uk−1

σ,i

)2]+∑

K∈M

|K|δt

(H(ρk

K

)−H

(ρk−1

K

))= −

d∑i=1

∑σ∈E(i)

S

Rkσ,i −

∑K∈M

RkK .

(3.23)Summing (3.23) for k = 1 to n, and using the fact that H′′(z) = γ zγ−2 for any γ ≥ 1 yields (3.21), with Rn

given by (3.22) and

C0 =12

d∑i=1

∑σ∈E(i)

S

|Dσ| ρ0σ |u0

σ,i|2 +∑

K∈M|K| H(ρ0

K).

Finally, the existence of a solution may be inferred by the Brouwer fixed point theorem, by an easy adap-tation of the proof of ([12], Prop. 5.2). This proof relies on the following set of mesh-dependent estimates: theconservativity of the mass balance discretization, together with the fact that the density is positive, yields anestimate for ρ in the L1-norm, and so, by a norm equivalence argument, of the pressure in any norm; the discretemomentum balance equation then provides a control on the velocity. Therefore, computing

(i) ρ from the mass balance for fixed u;(ii) p from ρ by the equation of state;(iii) and finally u from the momentum balance equation with fixed ρ and p.

yields an iteration in a bounded convex subset of a finite dimensional space. �

3.1.3. Passing to the limit in the scheme

The objective of this section is to show, in the one dimensional case, that if a sequence of solutions is controlledin suitable norms and converges to a limit, this latter necessarily satisfies a (part of the) weak formulation ofthe continuous problem.

The 1D version of the scheme which is studied in this section may be obtained from Scheme (3.1) by takingthe MAC variant, with only one horizontal stripe of grid cells, supposing that the vertical component of thevelocity (the degrees of freedom of which are located on the top and bottom boundaries) vanishes, and thatthe measure of the vertical faces is equal to 1. For the sake of readability, however, we completely rewrite this1D scheme, and, to this purpose, we first introduce some adaptations of the notations to the one dimensionalcase. For any K ∈ M, we denote by hK its length (so hK = |K|); when we write K = [σσ′], this means thateither K = (xσ , xσ′) or K = (xσ′ , xσ); if we need to specify the order, i.e. K = (xσ , xσ′) with xσ < xσ′ , thenwe write K = [

−→σσ′]. For an interface σ = K|L between two cells K and L, we define hσ = (hK + hL)/2, so, by

definition of the dual mesh, hσ = |Dσ|. If we need to specify the order of the cells K and L, say K is left of L,


then we write σ =−−→K|L. With these notations, the implicit scheme (3.1) may be written as follows in the one

dimensional setting:

∀K ∈ M, ρ0K =

1|K|

∫K

ρ0(x) dx,

∀σ ∈ Eint, u0σ =

1|Dσ|

∫Dσ

u0(x) dx,(3.24a)

∀K = [−→σσ′] ∈ M,

|K|δt

(ρn+1K − ρn

K) + Fn+1σ′ − Fn+1

σ = 0, (3.24b)

∀σ =−−→K|L ∈ Eint,

|Dσ|δt

(ρn+1

Dσun+1

σ − ρnDσun

σ

)+ Fn+1

L un+1L − Fn+1

K un+1K

− |Dσ| (ΔMu)n+1σ + pn+1

L − pn+1K = 0,

(3.24c)

∀K ∈ M, pn+1K = ℘

(ρn+1

K

)= (ρn+1

K )γ . (3.24d)

The mass flux in the discrete mass balance equation is given, for σ ∈ Eint, by:

Fn+1σ = ρn+1

σ un+1σ , (3.25)

where the upwind approximate density ρn+1σ at the face σ is defined by (3.3). In the momentum balance

equation, the application of the procedure described in Section 3.1.1 yields the following expression for thedensity associated with the dual cell Dσ with σ = K|L and for the mass fluxes at the dual face located at thecenter of the cell K = [σσ′]:

ρn+1Dσ

=1

2 |Dσ|(|K| ρn+1

K + |L| ρn+1L

), Fn+1

K =12(Fn+1

σ + Fn+1σ′), (3.26)

and the approximation of the velocity at this face is centered: un+1K = (un+1

σ + un+1σ′ )/2. Finally, for a face

σ =−−→K|L with K = [

−→σ′σ] and L = [

−−→σσ′′], the stabilization diffusion term reads:

−|Dσ| (ΔMu)n+1σ = ν

[1hK

(un+1σ − un+1

σ′ ) +1hL

(un+1

σ − un+1σ′′)]. (3.27)

Definition 3.5 (Regular sequence of discretizations). We define a regular sequence of discretizations(M(m), δt(m), ν(m))m∈N as a sequence of meshes, time steps and numerical diffusion coefficients satisfyingthe following assertions:

(i) both the time step δt(m) and the size h(m) of the mesh M(m), defined by h(m) = supK∈M(m) hK , tend tozero as m→ +∞;

(ii) there exists θ > 0 such that:

θ ≤ hK

hL≤ 1θ, ∀m ∈ N and K, L ∈ M(m) sharing a face,

(iii) the sequence of numerical diffusion coefficients (ν(m))m∈N is such that:

limm→+∞

ν(m) = 0, limm→+∞

(h(m))2

ν(m)= 0.


Let such a regular sequence of discretizations be given, and ρ(m), p(m) and u(m) be a solution given by thescheme (3.24) with the mesh M(m), the time step δt(m) and the numerical diffusion coefficient ν(m). To thediscrete unknowns, we associate piecewise constant functions on time intervals and on primal or dual meshes,so that the density ρ(m), the pressure p(m) and the velocity u(m) are defined almost everywhere on Ω × (0, T )by:

ρ(m)(x, t) =N−1∑n=0

∑K∈M

(ρ(m))n+1

KXK(x) X(n,n+1](t), p(m)(x, t) =

N−1∑n=0

∑K∈M

(p(m))n+1

KXK(x) X(n,n+1](t),

u(m)(x, t) =N−1∑n=0

∑σ∈E

(u(m))n+1

σXDσ (x) X(n,n+1](t),

where XK , XDσ and X(n,n+1] stand for the characteristic functions of the intervals K, Dσ and (tn, tn+1] respec-tively.

A weak solution to the continuous problem satisfies, for any ϕ ∈ C∞c

(Ω × [0, T )

):

−∫ T

0

∫Ω

[ρ ∂tϕ+ ρ u ∂xϕ] dxdt−∫

Ω

ρ0(x)ϕ(x, 0) dx = 0, (3.28a)

−∫ T

0

∫Ω

[ρ u ∂tϕ+ (ρ u2 + p) ∂xϕ

]dxdt−

∫Ω

ρ0(x)u0(x)ϕ(x, 0) dx = 0, (3.28b)

p = ργ . (3.28c)

Note that these relations are not sufficient to define a weak solution to the problem, since they do not implyanything about the boundary conditions. However, they allow to derive the Rankine–Hugoniot conditions; henceif we show that they are satisfied by the limit of a sequence of solutions to the discrete problem, this implies,loosely speaking, that the scheme computes correct shocks (i.e. shocks where the jumps of the unknowns andof the fluxes are linked to the shock speed by Rankine–Hugoniot conditions). This is the result we are seekingand which we state in Theorem 3.7. In order to prove this theorem, we need some definitions of interpolates ofregular test functions on the primal and dual mesh.

Definition 3.6 (Interpolates on one-dimensional meshes). Let Ω be an open bounded interval of R, let ϕ ∈C∞

c (Ω × [0, T )), and let M be a mesh of Ω. The interpolate ϕM of ϕ on the primal mesh M is defined by:

ϕM =N−1∑n=0

∑K∈M

ϕnK XK X[tn,tn+1),

where, for 0 ≤ n ≤ N and K ∈ M, we set ϕnK = ϕ(xK , t

n), with xK the mass center of K. The time and spacediscrete derivatives of the discrete function ϕM are defined by:

ðtϕM =N−1∑n=0

∑K∈M

ϕn+1K − ϕn

K

δtXK X[tn,tn+1), and ðxϕM =

N−1∑n=0

∑σ=

−−→K|L∈Eint

ϕnL − ϕn

K

hσXDσ X[tn,tn+1).

Let ϕE be an interpolate of ϕ on the dual mesh, defined by:

ϕE =N−1∑n=0

∑σ∈E

ϕnσ XDσ X[tn,tn+1),


where, for 1 ≤ n ≤ N and σ ∈ E , we set ϕnσ = ϕ(xσ , t

n), with xσ the abscissa of the interface σ. We also definethe time and space discrete derivatives of this discrete function by:

ðtϕE =N−1∑n=0

∑σ∈E

ϕn+1σ − ϕn

σ

δtXDσ X[tn,tn+1), and ðxϕE =

N−1∑n=0

∑K=[

−−→σσ′]∈M

ϕnσ′ − ϕn

σ

hKXK X[tn,tn+1).

Finally, we define ðxϕM,E by:

ðxϕM,E =N−1∑n=0

∑K=[

−−→σσ′ ]∈M

ϕnK − ϕn

σ

hK/2XDK,σ X[tn,tn+1),+

ϕnσ′ − ϕn

K

hK/2XDK,σ′ X[tn,tn+1).

Theorem 3.7 (Consistency of the one-dimensional implicit scheme). Let Ω be an open bounded inter-val of R. We suppose that the initial data satisfies ρ0 ∈ L∞(Ω), 1/ρ0 ∈ L∞(Ω) and u0 ∈ L∞(Ω).Let (M(m), δt(m), ν(m))m∈N be a regular sequence of discretizations in the sense of Definition 3.5, and(ρ(m), p(m), u(m))m∈N be the corresponding sequence of solutions. We suppose that this sequence converges inLp(Ω × (0, T ))3, for 1 ≤ p < +∞, to (ρ, p, u) ∈ L∞(Ω × (0, T ))3. We suppose in addition that both sequences(ρ(m))m∈N and (1/ρ(m))m∈N are uniformly bounded in L∞(Ω × (0, T )).

Then the limit (ρ, p, u) satisfies the system (3.28).

Proof. With the assumed convergence for the sequence of solutions, the limit clearly satisfies the equation ofstate (note that in reality this is the difficult point to prove with the estimates at hand, see e.g. [12]). Theproof of this theorem is thus obtained by passing to the limit in the scheme, first for the mass balance equationand then for the momentum balance equation. Thanks to the assumption on the initial condition, the stabilityestimate of Proposition 3.3 is uniform with respect to m, and thus provides uniform bounds for some spacetranslates of the solution (see the expression (3.22) of the remainder term), which are used all along the proof.In particular, using in addition the assumption that both sequences (ρ(m))m∈N and (1/ρ(m))m∈N are uniformlybounded in L∞(Ω × (0, T )), exploiting the last part of the remainder term, we get the following weak BVestimate for ρ:

∀m ∈ N,

N(m)−1∑n=0

δt∑

σ=K|L∈E(m)int

∣∣∣∣(u(m))n+1

σ

∣∣∣∣ [(ρ(m))n+1

K−(ρ(m))n+1

L

]2≤ C, (3.29)

where C stands for a real number which is independent of m.

Mass balance equation – Let ϕ ∈ C∞c (Ω × [0, T )). Let m ∈ N, M(m), δt(m) and ν(m) be given. Dropping

for short the superscript (m), let ϕM be the interpolate of ϕ on the primal mesh and let ðtϕM and ðxϕMbe its time and space discrete derivatives in the sense of Definition 3.6. Thanks to the regularity of ϕ, thesefunctions respectively converge in Lr(Ω× (0, T )), for r ≥ 1 (including r = +∞), to ϕ, ∂tϕ and ∂xϕ respectively.In addition, ϕM(m)(·, 0) converges to ϕ(·, 0) in Lr(Ω) for r ≥ 1 as m→ +∞. Since the support of ϕ is compactin Ω × [0, T ), for m large enough, ϕM(m) vanishes at the boundary cells and at the final time; hereafter, wesystematically assume that this indeed the case.

Let us multiply the discrete mass balance equation (3.24b) by δt ϕnK , and sum the result on n ∈ {0, . . . , N−1}

and K ∈ M, to obtain T (m)1 + T

(m)2 = 0 with

T(m)1 =

N−1∑n=0

∑K∈M

|K| (ρn+1K − ρn

K) ϕnK , T

(m)2 =

N−1∑n=0

δt∑

K=[−−→σσ′]∈M

[Fn+1

σ′ − Fn+1σ

]ϕn

K .


(In the above expressions and in the remainder of the proof, we whall omit the superscript (m) in the summationsfor the sake of clarity.) Reordering the sums in T

(m)1 yields:

T(m)1 = −

N−1∑n=0

δt∑

K∈M|K| ρn+1

K

ϕn+1K − ϕn

K

δt−∑

K∈M|K| ρ0

K ϕ0K ,

= −∫ T

0

∫Ω

ρ(m)ðt ϕM(m) dxdt−

∫Ω

ρ0(x) ϕM(m)(x, 0) dx.

Since, by assumption, the sequence of discrete solutions converges in Lr(Ω × (0, T )) for r ≥ 1, we get:

limm−→+∞

T(m)1 = −

∫ T

0

∫Ω

ρ ∂tϕdxdt−∫

Ω

ρ0(x) ϕ(x, 0) dx.

Reordering the sums in T (m)2 , we get:

T(m)2 = −

N−1∑n=0

δt∑

σ=−−→K|L∈Eint

|Dσ| ρn+1σ un+1

σ

ϕnL − ϕn

K

hσ,

where hσ (which is equal to |Dσ|) is by definition equal to |xL − xK | and we recall that ρn+1σ is the upwind

approximation of ρn+1 at the face σ. Using the fact that |Dσ| = (|K|+ |L|)/2, we may write T (m)2 = T (m)

2 +R(m)2

with:

T (m)2 = −

N−1∑n=0

δt∑


[|K|2

ρn+1K +

|L|2

ρn+1L

]un+1

σ

ϕnL − ϕn

K

hσ,

R(m)2 = −

N−1∑n=0

δt∑


[|K|2(ρn+1

K − ρn+1L

)(un+1

σ )− +|L|2(ρn+1

K − ρn+1L

)(un+1

σ )+]ϕn

L − ϕnK

hσ·

Therefore we get

T (m)2 = −

∫ T

0

∫Ω

ρ(m) u(m)ðxϕM(m) dxdt, and lim

m−→+∞T (m)

2 = −∫ T

0

∫Ω

ρ u ∂xϕdxdt,

and the remainder term R(m)2 is bounded as follows:

|R(m)2 | ≤ Cϕ

N−1∑n=0

δt∑

σ=K|L∈Eint

|Dσ| |ρn+1K − ρn+1

L | |un+1σ |

≤ Cϕ h1/2N−1∑n=0

δt

⎡⎣ ∑σ=K|L∈Eint

∣∣un+1σ

∣∣ ∣∣ρn+1K − ρn+1

L

∣∣2⎤⎦1/2 ⎡⎣ ∑σ=K|L∈Eint

|Dσ|∣∣un+1

σ

∣∣⎤⎦1/2

,

where the notation Cϕ means that this real number only depends on the function ϕ. Thanks to the stabilityestimate (3.29), this term tends to zero when m tends to +∞.Momentum balance equation – Let ϕE , ðtϕE and ðxϕE be the interpolate of ϕ on the dual mesh and itsdiscrete time and space derivatives, in the sense of Definition 3.6, which converge in Lr(Ω × (0, T )), for r ≥ 1(including r = +∞), to ϕ, ∂tϕ and ∂xϕ respectively.


Let us multiply the discrete momentum balance equation (3.24c) by δt ϕnσ, and sum the result over n ∈

{0, . . . , N − 1} and σ ∈ Eint. We obtain T (m)1 + T

(m)2 + T

(m)3 + T

(m)4 = 0 with:

T(m)1 =

N−1∑n=0

∑σ∈Eint

|Dσ|(ρn+1

Dσun+1

σ − ρnDσun

σ

)ϕn

σ ,

T(m)2 =

N−1∑n=0

δt∑


[Fn+1

L un+1L − Fn+1

K un+1K

]ϕn

σ ,

T(m)3 =

N−1∑n=0

δt∑


(pn+1L − pn+1

K ) ϕnσ ,

T(m)4 =

N−1∑n=0

δt∑

σ∈Eint

⎡⎣ ∑K=[σσ′ ]

ν

hK

(un+1

σ − un+1σ′)⎤⎦ ϕn

σ.

Thanks to the definition (3.26) of the density on the dual mesh ρDσ , reordering the sums, we get for T (m)1 :

T(m)1 = −

N−1∑n=0

δt∑

σ=K|L∈Eint

[|K|2ρn+1

K +|L|2ρn+1

L

]un+1

σ

ϕn+1σ − ϕn

σ

δt−

∑σ=K|L∈Eint

[|K|2ρ0

K +|L|2ρ0

L

]u0

σ ϕ0σ.

Therefore:

T(m)1 = −

∫ T

0

∫Ω

ρ(m) u(m)ðt ϕM(m) dxdt−

∫Ω

(ρ(m))0(x) (u(m))0(x) ϕM(m)(x, 0) dx.

From the definition (3.24a) of the initial conditions and the assumed regularity of ρ0 and u0, the sequences((ρ(m))0

)and(u(m))0

)converge in Lr(Ω), for 1 ≤ r < +∞, to ρ0 and u0 respectively. From the convergence

assumption for the sequence of discrete solutions, we thus get:

limm−→+∞

T(m)1 = −

∫ T

0

∫Ω

ρ u ∂tϕdxdt −∫

Ω

ρ0(x) u0(x) ϕ(x, 0) dx.

Let us now turn to T (m)2 . From the expression (3.26) of the fluxes FK and the values uK , reordering the sums,

we get:

T(m)2 = −1

4

N−1∑n=0

δt∑

K=[−−→σσ′]∈M

(ρn+1

σ un+1σ + ρn+1

σ′ un+1σ′) (

un+1σ + un+1

σ′)

(ϕnσ′ − ϕn

σ) ,

which we write T (m)2 = T (m)

2 + R(m)2 with:

T (m)2 = −1

2

N−1∑n=0

δt∑

K=[−−→σσ′]∈M

ρn+1K

[(un+1

σ

)2+ (un+1

σ′ )2]

(ϕnσ′ − ϕn

σ) .


This term reads:

T (m)2 = −

∫ T

0

∫Ω

ρ(m) (u(m))2 ðxϕE dxdt, and so limm−→+∞

T (m)2 = −

∫ T

0

∫Ω

ρ u2 ∂xϕdxdt.

The remainder term R(m)2 reads:

R(m)2 = −1

4

N−1∑n=0

δt∑

K=[−−→σσ′]∈M

[(ρn+1

σ un+1σ + ρn+1

σ′ un+1σ′ )(un+1

σ +un+1σ′ )− 2ρn+1

K

((un+1

σ

)2+(un+1

σ′)2)]

(ϕnσ′ −ϕn

σ).

Expanding the quantity 2 ρn+1K ((un+1

σ )2 + (un+1σ′ )2) thanks to the identity 2(a2 + b2) = (a + b)2 + (a − b)2, we

get R(m)2 = R(m)

2,1 + R(m)2,2 :

R(m)2,1 = −1

4

N−1∑n=0

δt∑

K=[−−→σσ′]∈M

[((ρn+1

σ − ρn+1K ) un+1

σ + (ρn+1σ′ − ρn+1

K ) un+1σ′

)(un+1

σ + un+1σ′ )]

(ϕnσ′ − ϕn

σ),

R(m)2,2 =

14

N−1∑n=0

δt∑

K=[−−→σσ′]∈M

ρn+1K (un+1

σ − un+1σ′ )2 (ϕn

σ′ − ϕnσ).

First we study R(m)2,1 . Thanks to the definition (3.3) of the upwind value ρn+1

σ , reordering the sum by faces, weget that:

|R(m)2,1 | =

14

∣∣N−1∑n=0

δt∑

σ∈Eint,

σ=L→K, K=[σσ′]

(ρn+1L − ρn+1

K ) un+1σ (un+1

σ + un+1σ′ ) (ϕn

σ − ϕnσ′ )∣∣,

where the notation L→ K means that the flow is going from L to K, or, in other words, that if un+1σ ≥ 0 (resp.

un+1σ ≤ 0), the cells K and L are chosen such that σ =

−−→L|K (resp. σ =

−−→K|L). Since |ϕn

σ − ϕnσ′ | ≤ Cϕ |K| ≤

Cϕ (|Dσ| + |Dσ′ |), we get:

|R(m)2,1 | ≤ Cϕ

4

N−1∑n=0

δt∑

σ∈Eint,


(|Dσ| + |Dσ′ |

)|ρn+1

L − ρn+1K | |un+1

σ | |un+1σ + un+1

σ′ |.

Therefore, by the Cauchy−Schwarz inequality, we get:

|R(m)2,1 | ≤ Cϕ

4h1/2

N−1∑n=0

δt


|un+1σ | (ρn+1

L − ρn+1K )2

⎤⎦1/2

×

⎡⎢⎢⎢⎣ ∑σ∈Eint,


(|Dσ| + |Dσ′ |

)|un+1

σ |(un+1

σ + un+1σ′)2⎤⎥⎥⎥⎦

1/2

. (3.30)

Since the ratio of the size of two neighboring cells is bounded by the regularity assumption on the mesh (item(ii) of Def. 3.5), we get from the estimate (3.29) on the solution:

|R(m)2,1 | ≤ C h1/2 ‖u(m)‖3/2

L3(Ω×(0,T )), (3.31)


where the real number C is independent of m and therefore R(m)2,1 tends to zero when m tends to +∞. For

R(m)2,2 , we have, thanks to the estimate (3.21):

|R(m)2,2 | ≤ Cϕ h2

N−1∑n=0

δt∑

K∈Mρn+1

K

1hK

(un+1σ − un+1

σ′ )2 ≤ Ch2

ν(m)‖ρ(m)‖L∞(Ω×(0,T )),

where C > 0 does not depend on m; therefore, this term also tends to zero when m tends to +∞, since, byassumption, h2/ν(m) tends to zero.

We turn to the term T(m)3 :

T(m)3 = −

N−1∑n=0

δt∑

K=[−−→σσ′]∈M

|K| pn+1K

ϕnσ′ − ϕn

σ

hK= −

∫ T

0

∫Ω

p(m)ðxϕE dxdt,

and therefore,

limm−→+∞

T(m)3 = −

∫ T

0

∫Ω

p ∂xϕdxdt.

Let us finally study T (m)4 . Reordering the sums, we get:

T(m)4 =

N−1∑n=0

δt∑

K=[−−→σσ′]∈M

ν(m)

hK(un+1

σ − un+1σ′ ) (ϕn

σ − ϕnσ′ ).

The Cauchy−Schwarz inequality yields:

|T (m)4 | ≤

⎡⎢⎣N−1∑n=0

δt∑

K=[−−→σσ′ ]∈M

ν(m)

hK(un+1

σ − un+1σ′ )2

⎤⎥⎦1/2 ⎡⎢⎣N−1∑

n=0

δt∑

K=[−−→σσ′]∈M

ν(m)

hK(ϕn

σ − ϕnσ′ )2

⎤⎥⎦1/2

,

and thus, in view of the estimate (3.21), this term tends to zero when ν(m) tends to zero.

Conclusion – Gathering the limits of all the terms of the mass and momentum balance equations concludesthe proof. �

Remark 3.8 (Sharper bounds and convergence assumptions). The convergence properties and bounds assumedfor the solution have been chosen so as to match what may be observed in the theoretical works on compressibleNavier–Stokes equations [13,44,53]. Note that they are weaker than the assumptions of the Lax–Wendroff theo-rem for colocated finite volume schemes on hyperbolic systems, see e.g. [11], Theorem 5.3. However, examiningthe proof of Theorem 3.7, we observe that what we really need is that the sequences ρ(m)u(m), ρ(m)(u(m))2,p(m)u(m) converge in the distribution sense to ρu, ρu2 and pu respectively, that (ρ(m))γ converge a.e. to ργ , andthat the sequence (u(m))m∈N be bounded in L3(Ω × (0, T )). The required second assumption for (ν(m))m∈N isin fact:

limm→+∞

(h(m))2

ν(m)‖ρ(m)‖L∞(Ω×(0,T )) = 0,

and may be verified, for instance supposing a relation between δt(m) and h(m) and invoking inverse inequalities,with milder estimates on (ρ(m))m∈N. Finally, the bound of (1/ρ(m))m∈N in L∞(Ω × (0, T )) (which, looselyspeaking, means that the appearance of void is excluded) is needed to obtain the weak-BV estimate:

limm→+∞

h(m)N∑

n=1

∑σ=K|L∈Eint

|unσ| (ρn

K − ρnL)2 = 0 (3.32)


from the “weighted weak-BV estimate” (3.21):

limm→+∞

h(m)N∑

n=1

∑σ=K|L∈Eint

(ρnσ,γ)γ−2 |un

σ| (ρnK − ρn

L)2 = 0,

where we recall that ρnσ,γ is equal to either ρn

K or ρnL. This assumption is thus useless for γ ≤ 2. For γ > 2, in

the case of a non-vanishing viscosity, equation (3.32) may perhaps be derived by using the density itself as testfunction in the discrete mass balance equation, and invoking a control of the divergence of the velocity (fromthe momentum balance diffusion term), see ([12], Prop. 5.5) for such a computation in the steady case.

Remark 3.9 (Less sharp bounds and more general meshes). The assumption that the ratio of the size of twoneighboring cells is bounded, i.e. assumption (ii) of Definition 3.5, is only used to derive (3.31) from (3.30),which allows to conclude that the remainder term R(m)

2,1 tends to zero invoking a control of the velocity only inL3(Ω×(0, T )). If we choose to use a uniform bound in L∞(Ω×(0, T )) for the sequence of approximate solutions,we may replace (3.31) by ∣∣∣R(m)

2,1

∣∣∣ ≤ C h1/2 ‖u(m)‖3/2

L∞(Ω×(0,T )),

and assumption (ii) is useless.

We now turn to the entropy condition. Let us first recall that η = ρEk +H(ρ) is an entropy of the continuousproblem (1.1), in the sense that if we sum the formal kinetic energy (3.13) and elastic potential balance (3.18),we get:

∂tη + ∂x

((η + p)u

)= 0. (3.33)

In fact, in order to avoid to invoke unrealistic regularity assumption, such a computation should be done onregularized equations (obtained by adding diffusion perturbation terms), and, when making these regularizationterms tend to zero, positive measures appear at the left-hand-side of (3.33), so that we get in the distributionsense:

∂tη + ∂x

((η + p)u

)≤ 0. (3.34)

An entropy solution to (1.1) is thus required to satisfy, for any ϕ ∈ C∞c

(Ω × [0, T )

), ϕ ≥ 0:∫ T

0

∫Ω

[η ∂tϕ+ (η + p)u ∂xϕ

]dx dt+

∫Ω

η0(x) ϕ(x, 0) dx ≥ 0, (3.35)

where η0 =12ρ0 u

20 + H(ρ0).

Theorem 3.10 (Entropy consistency, implicit scheme). Under the assumptions of Theorem 3.7, (ρ, p, u) sat-isfies the entropy condition (3.35).

Proof. Let ϕ ∈ C∞c

(Ω× [0, T )

), ϕ ≥ 0. Using the notations introduced in Definition 3.6, we multiply the kinetic

balance equation (3.14) by ϕnσ, and the elastic potential balance (3.19) by ϕn

K , sum over the faces and cellsrespectively, to get ∑

E∈Eint

T n+1σ ϕn

σ +∑

K∈MT n+1

K ϕnK = −

∑E∈Eint

Rn+1σ ϕn

σ −∑

K∈MRn+1

K ϕnK , (3.36)

where, for σ =−−→K|L, K = [

−→σ′σ] and L = [

−−→σσ′′],

T n+1σ =

12|Dσ|δt

[ρn+1

Dσ(un+1

σ )2 − ρnDσ

(unσ)2]

+12Fn+1

L un+1σ un+1

σ′′ − 12Fn+1

K un+1σ un+1

σ′ + (pn+1L − pn+1

K ) un+1σ ,


for K = [−→σσ′],

T n+1K =

|K|δt

[H(ρn+1

K ) −H(ρnK)]

+ un+1σ′ H(ρn+1

σ′ ) − un+1σ H(ρn+1

σ ) + pn+1K (un+1

σ′ − un+1σ ),

and the quantities Rn+1σ and Rn+1

K are given by (the one-dimensional version of) equations (3.15) and (3.20)respectively.

The discrete weak form of the entropy balance is obtained by integrating in time (i.e. summing over the timesteps) equation (3.36). We obtain T (m)

1 + T(m)2 + T

(m)3 + T

(m)4 + T

(m)5 +R(m) = 0 with:

T(m)1 =

12

N−1∑n=0

∑σ∈Eint

|Dσ|[ρn+1

Dσ(un+1

σ )2 − ρnDσ

(unσ)2]ϕn

σ, (3.37a)

T(m)2 =

12

N−1∑n=0

∑K∈M

|K|[H(ρn+1

K ) −H(ρnK)]ϕn

K , (3.37b)

T(m)3 =

12

N−1∑n=0

δt∑

σ=−−→K|L∈Eint,

K=[−−→σ′σ], L=[

−−→σσ′′]

[Fn+1

L un+1σ un+1

σ′′ − Fn+1K un+1

σ un+1σ′]ϕn

σ, (3.37c)

T(m)4 =

N−1∑n=0

δt∑

K=[−−→σσ′]∈M

[un+1

σ′ H(ρn+1σ′ ) − un+1

σ H(ρn+1σ )]ϕn

K , (3.37d)

T(m)5 =

12

N−1∑n=0

δt

⎡⎢⎣ ∑σ=

−−→K|L∈Eint

(pn+1L − pn+1

K ) un+1σ ϕn

σ +∑

K=[−−→σσ′]∈M

pn+1K (un+1

σ′ − un+1σ ) ϕn

K

⎤⎥⎦ , (3.37e)

R(m) =12

N−1∑n=0

δt

⎡⎢⎣ ∑σ=

−−→K|L∈Eint

Rnσ ϕn

σ +∑

K=[−−→σσ′ ]∈M

RnK ϕn

K

⎤⎥⎦. (3.37f)

We first study T(m)1 . Reordering the summations and then using the definition (3.26) of the density at the

faces, we get:

T(m)1 = − 1

2

N−1∑n=0

δt∑σ∈E

|Dσ| ρn+1Dσ

(un+1σ )2

ϕn+1σ − ϕn

σ

δt− 1

2

∑σ∈E

|Dσ|ρ0Dσ

(u0σ)2 ϕ0

σ

= − 12

∫ T

0

∫Ω

ρ(m) (u(m))2 ðtϕE dxdt− 12

∫Ω

(ρ(m))0(x)[(u(m))0(x)

]2ϕE(x, 0) dx.

By the definition (3.53a) of the initial conditions of the scheme, since both ρ0 and u0 are supposed to belongto L∞(Ω), (ρ(m))0 and (u(m))0 converge to ρ0 and u0 respectively in Lr(Ω), for r ≥ 1. Since, by assumption,the sequence of discrete solutions converges in Lr(Ω× (0, T )) for r ≥ 1, we can pass to the limit in the previousrelation, to get:

limm−→+∞

T(m)1 = −1

2

∫ T

0

∫Ω

ρ (u)2 ∂tϕdxdt− 12

∫Ω

ρ0(x)u0(x)2 ϕ(x, 0) dx. (3.38)


By a similar computation, we get for T (m)2 :

T(m)2 = −

N−1∑n=0

δt∑

K∈M|K| H(ρn+1

K )ϕn+1

K − ϕnK

δt−∑σ∈E

|K| H(ρ0K) ϕ0

K

= −∫ T

0

∫Ω

H(ρ(m)) ðtϕM(m) dxdt−∫

Ω

H((ρ(m))0

)(x) ϕM(m)(x, 0) dx,

and therefore:

limm−→+∞

T(m)2 = −

∫ T

0

∫Ω

H(ρ) ∂tϕdxdt−∫

Ω

H(ρ0)(x)ϕ(x, 0) dx. (3.39)

Let us now study the kinetic energy convection term T(m)3 which reads, after reordering the summations:

T(m)3 = −1

2

N−1∑n=0

δt∑

K=[−−→σσ′]∈M

Fn+1K un+1

σ un+1σ′ (ϕn

σ − ϕnσ′) .

Using now the definition of the mass fluxes at the dual edges, we have:

T(m)3 = −1

4

N−1∑n=0

δt∑

K=[−−→σσ′]∈M

(ρn+1

σ un+1σ + ρn+1

σ′ un+1σ′ ) un+1

σ′ un+1σ

(ϕn

σ − ϕnσ′).

We now split T (m)3 = T (m)

3 + R(m)3 , where

T (m)3 = −1

4

N−1∑n=0

δt∑

K=[−−→σσ′]∈M

ρn+1K

[(un+1

σ )3 + (un+1σ′ )3

] (ϕn

σ − ϕnσ′)

= −12

∫ T

0

∫Ω

ρ(m)(u(m))3ðxϕE dxdt,

so that

limm−→+∞

T (m)3 = −1

2

∫ T

0

∫Ω

ρu3∂xϕdxdt,

and

R(m)3 = −1

4

N−1∑n=0

δt∑

K=[−−→σσ′]∈M

[(ρn+1

σ un+1σ + ρn+1

σ′ un+1σ′ ) un+1

σ un+1σ′ − ρn+1

K

((un+1

σ )3 + (un+1σ′ )3

)](ϕn

σ − ϕnσ′).

Expanding the quantity (un+1σ )3 + (un+1

σ′ )3 thanks to the identity a3 + b3 = (a + b)(ab + (a − b)2), we obtainR(m)

3 = R(m)3,1 + R(m)

3,2 with:

R(m)3,1 = −1

4

N−1∑n=0

δt∑

K=[−−→σσ′]∈M

[(ρn+1

σ − ρn+1K )un+1

σ + (ρn+1σ′ − ρn+1

K )un+1σ′

]un+1

σ un+1σ′ (ϕn

σ − ϕnσ′ ),

R(m)3,2 =

14

N∑n=0

δt∑

K=[−−→σσ′]∈M

ρn+1K (un+1

σ + un+1σ′ ) (un+1

σ − un+1σ′ )2 (ϕn

σ − ϕnσ′).

Reordering the sums, the term R(m)3,1 reads:

R(m)3,1 =

14

N−1∑n=0

δt∑

σ∈Eint,


εn+1σ (ρn+1

L − ρn+1K ) (un+1

σ )2 un+1σ′ (ϕn

σ − ϕnσ′),


where εn+1σ = ±1 and the notation L → K means that the flow is going from L to K. Thanks to the

Cauchy−Schwarz inequality, we get, by the regularity of ϕ:

|R(m)3,1 | ≤ Cϕ h1/2

⎡⎣N−1∑n=0

δt∑

σ=K|L∈Eint

|un+1σ | (ρn+1

L − ρn+1K )2

⎤⎦1/2

×

⎡⎢⎢⎢⎣N−1∑n=0

δt∑

σ∈Eint,


|K| |un+1σ | (un+1

σ un+1σ′ )2

⎤⎥⎥⎥⎦1/2

,

and thus:|R(m)

3,1 | ≤ Cϕ h1/2 ‖u(m)‖5/2

L5(Ω×(0,T )).

We now turn to R(m)3,2 . Thanks to the regularity of ϕ, we get:

|R(m)3,2 | ≤ Cϕ

(h(m))2

ν(m)‖ρ(m)‖L∞(Ω×(0,T )) ‖u(m)‖2

L∞(Ω×(0,T ))

∑K=[σσ′]∈M

ν(m)

hK(un+1

σ − un+1σ′ )2,

and thus R(m)3,2 also tends to zero when m tends to +∞ as soon as the ratio (h(m))2/ν(m) tends to zero. As a

consequence, we get that

limm−→+∞

T(m)3 = −1

2

∫ T

0

∫Ω

ρu3∂xϕdxdt.

Expressing the mass fluxes as a function of the unknowns in T (m)4 and reordering the sums, we get:

T(m)4 = −

N−1∑n=0

δt∑


H(ρn+1σ )un+1

σ (ϕnK − ϕn

L).

Let us write T (m)4 = T (m)

4 + R(m)4 , with, thanks to the definition of the upwind density (3.3) at the face:

T (m)4 = −

N−1∑n=0

δt∑


[|DK,σ|H(ρn+1

K ) + |DL,σ|H(ρn+1L )]un+1

σ

ϕnK − ϕn

L

hσ,

R(m)4 =

N−1∑n=0

δt∑


|DL,σ|[H(ρn+1

L ) −H(ρn+1K )]un+1

σ

ϕnK − ϕn

L

hσ·

We have:

T (m)4 = −

∫ T

0

∫Ω

H(ρ(m))u(m)ðxϕM(m) dxdt, so lim

m−→+∞T (m)

4 = −∫ T

0

∫Ω

H(ρ) u ∂xϕdxdt.

By the regularity of ϕ, we get:

|R(m)4 | ≤ Cϕ h(m)

N−1∑n=0

δt∑

σ=K|L∈Eint

∣∣H(ρn+1K ) −H(ρn+1

L )∣∣ |un+1

σ |.


Since both sequences (ρ(m))m∈N and (1/ρ(m))m∈N are supposed to be uniformly bounded, we have∣∣H(ρn+1

K ) −H(ρn+1

L )∣∣ ≤ C |ρn+1

K −ρn+1L | with a constant real number C, and therefore R(m)

4 tends to zero as h(m). Therefore,

limm→+∞

T(m)4 = −

∫ T

0

∫Ω

H(ρ) u ∂xϕdxdt. (3.40)

Reordering the sums in the term T(m)5 , we obtain:

T(m)5 =

N−1∑n=0

−δt∑

K=[−−→σσ′]∈M

pn+1K (un+1

σ′ ϕnσ′ − un+1

σ ϕnσ) + pn+1

K (un+1σ − un+1

σ′ ) ϕnK ,

hence:

T(m)5 = −

N−1∑n=0

δt∑

K=[−−→σσ′ ]∈M

|K|2pn+1

K un+1σ

ϕnK − ϕn

σ

hK/2+

|K|2pn+1

K un+1σ′

ϕnσ′ − ϕn

K

hK/2

= −∫ T

0

∫Ω

p(m) u(m)ðxϕM,E dxdt.

and so, since ðxϕM,E converges to ∂xϕ in Lr(Ω), for r ≥ 1:

limm−→+∞

T(m)5 = −

∫ T

0

∫Ω

p u ∂xϕdxdt. (3.41)

From (3.38)−(3.41), we get that

limm−→+∞

5∑i=1

T(m)i = −

∫ T

0

∫Ω

[η ∂tϕ+ (η + p)u ∂xϕ

]dxdt−

∫Ω

η0(x) ϕ(x, 0) dx.

In order to complete the proof of Theorem 3.17, there only remains to show that limm→+∞R(m) ≥ 0. Since weonly seek an inequality, the non-negative part of R(m), i.e. the first part in Rn+1

σ and the whole term Rn+1K ,

poses no problem, and we only have to study the terms coming from the second part of Rn+1σ , which reads:

(Rdiff)n+1σ =

⎡⎣ ∑K=[σσ′]

ν(m)

hK(un+1

σ − un+1σ′ )

⎤⎦ un+1σ .

For 0 ≤ n ≤ N − 1 and K ∈ M, K = [σσ′], let us define the quantity Qn+1K by:

Qn+1K =

ν(m)

hK(un+1

σ − un+1σ′ )2.

We have Qn+1K ≥ 0, and, reordering the summation, we get that

Q(m) =

∣∣∣∣∣N−1∑n=0

δt

[ ∑σ∈Eint

ϕnσ (Rdiff)n+1

σ −∑

K∈Mϕn

K Qn+1K

]∣∣∣∣∣=

∣∣∣∣∣∣∣N−1∑n=0

δt∑

K=[−−→σσ′]∈M

ν

hK(un+1

σ − un+1σ′ )

[un+1

σ (ϕσ − ϕK) − un+1σ′ (ϕσ′ − ϕK)

]∣∣∣∣∣∣∣ .


By the Cauchy−Schwarz inequality and the regularity of ϕ, we thus get:

Q(m) ≤Cϕ

⎡⎢⎣N−1∑n=0

δt∑

K=[−−→σσ′]∈M

ν(m)

hK(un+1

σ − un+1σ′ )2

⎤⎥⎦1/2

×

⎡⎢⎣N−1∑n=0

δt∑

K=[−−→σσ′]∈M

ν(m)(|DK,σ| (un+1

σ )2 + |DK,σ′ | (un+1σ′ )2

)⎤⎥⎦1/2

.

From estimate (3.21), we thus get thatQ(m) ≤ C (ν(m))1/2,

where C > 0 only depends on ϕ, on C0 and on the assumed bounds on the solution. Since ν(m) tends to 0 withm, we then obtain that limm→+∞R(m) ≥ 0, which concludes the proof. �

3.2. The pressure correction scheme for the barotropic equations

The implicit scheme which we studied in the previous section is easy to write, but difficult to implement inpractice, because of the large nonlinear systems to be solved at the algebraic level. Pressure correction methodsare based on the idea that one may compute the velocity and the pressure in a sequential way, thus yieldinga more practical scheme. The algorithm presented in this section is implemented in the open-source softwarecomponent library for fluid flows simulation CALIF3S [5], developed at IRSN on the basis of the softwarecomponents library PELICANS [55]; in this context, it is routinely used for industrial applications.

3.2.1. The scheme

In the algorithm given below, the velocity is predicted by solving the momentum balance equation with aknown pressure. This latter is obtained from the beginning-of-step pressure through a “renormalization” step,in order to be able to perform the stability analysis (stability of the scheme and satisfaction of the entropycondition). Note that the renormalization proposed here is different than that proposed in [22] in the contextof variable density incompressible flows or in [14] in the context of compressible barotropic flows. Indeed, inthese latter works, this step requires the resolution of a discrete elliptic problem for the pressure, while, here,we only scale the pressure gradient by a simple weight. Then, the velocity is corrected and the other variablesare advanced in time. As for the implicit scheme, a discrete kinetic energy balance can be derived, providedthat the mass balance over the dual cells (3.5) holds; since the mass balance is not yet solved when performingthe prediction step, this leads us to perform a time shift of the density at this stage. The algorithm reads, for0 ≤ n ≤ N − 1:

Pressure gradient renormalization step:

∀σ ∈ E ,(∇p)n+1

σ=

√ρn

Dσ

ρn−1Dσ

(∇p)nσ . (3.42a)

Prediction step – Solve for un+1:

For 1 ≤ i ≤ d, ∀σ ∈ E(i)S ,

|Dσ|δt

(ρnDσun+1

σ,i − ρn−1Dσ

unσ,i) +

∑ε∈E(Dσ)

Fnσ,εu

n+1ε,i − |Dσ| (ΔMu)n+1

σ,i + |Dσ| (∇p)n+1σ,i = 0. (3.42b)


Correction step – Solve for ρn+1, pn+1 and un+1:

For 1 ≤ i ≤ d, ∀σ ∈ E(i)S ,

|Dσ|δt

ρnDσ

(un+1σ,i − un+1

σ,i ) + |Dσ|[(∇p)n+1

σ,i − (∇p)n+1σ,i

]= 0, (3.42c)

∀K ∈ M,|K|δt

(ρn+1K − ρn

K) +∑

σ∈E(K)

Fn+1K,σ = 0, with Fn+1

K,σ = |σ| ρn+1σ un+1

K,σ , (3.42d)

∀K ∈ M, pn+1K = (ρn+1

K )γ . (3.42e)

Recall that the notation ρn+1σ in (3.42d) stands for the upwind choice of ρ defined by (3.3), while ρn

Dσ

(resp. ρn−1Dσ

) in (3.42b) is the convex combination of ρnK and ρn

L (resp. ρn−1K and ρn−1

L ) defined by (3.4). Theinitialization of the scheme is performed as follows. First, ρ−1 and u0 are given by the average of the initialconditions ρ0 and u0 on the primal and dual cells respectively:

∀K ∈ M, ρ−1K =

1|K|

∫K

ρ0(x) dx,

for 1 ≤ i ≤ d, ∀σ ∈ E(i)S , u0

σ,i =1

|Dσ|

∫Dσ

(u0(x))i dx.

(3.43)

Then, we compute ρ0 by solving the mass balance equation (3.42d). Finally, the initial pressure p0 is computedfrom the initial density ρ0 by the equation of state: ∀K ∈ M, p0

K = (ρ0K)γ . This procedure allows to perform

the first prediction step with (ρ−1Dσ

)σ∈E , (ρ0Dσ

)σ∈E and the dual mass fluxes satisfying the mass balance.

3.2.2. Estimates

As for the implicit scheme, we begin with an estimate on the velocity which is a discrete equivalent of thekinetic energy balance.

Lemma 3.11 (Discrete kinetic energy balance, pressure correction scheme). A solution to the system (3.42)satisfies the following equality, for 1 ≤ i ≤ d, σ ∈ E(i)

S and 0 ≤ n ≤ N − 1:

12|Dσ|δt

[ρn

Dσ(un+1

σ,i )2−ρn−1Dσ

(unσ,i)

2]+

12

∑ε=Dσ|Dσ′

Fnσ,ε u

n+1σ,i un+1

σ′,i + |Dσ| (∇p)n+1σ,i un+1

σ,i = −Rn+1σ,i −Pn+1

σ,i , (3.44)

where

Rn+1σ,i =

|Dσ|2 δt

ρn−1Dσ

(un+1

σ,i − unσ,i

)2 +

⎡⎣ ∑ε=Dσ |Dσ′

ν hd−2ε (un+1

σ,i − un+1σ′,i )

⎤⎦ un+1σ,i ,

Pn+1σ,i =

|Dσ| δt2 ρn

Dσ

[((∇p)n+1

σ,i

)2 − ((∇p)n+1σ,i

)2].

(3.45)

Proof. Let us multiply the velocity prediction equation (3.42b) by the corresponding velocity unknown un+1σ,i ,

and use the equality (A.9) of Lemma A.2, on the dual mesh and with P = Dσ. We obtain:

12|Dσ|δt

[ρn

Dσ(un+1

σ,i )2 − ρn−1Dσ

(unσ,i)

2]

+12

∑ε=Dσ|Dσ′

Fnσ,ε u

n+1σ,i un+1

σ′,i

+12

|Dσ|δt

ρn−1Dσ

(un+1

σ,i − unσ,i

)2 − |Dσ|(ΔMu)n+1σ,i un+1

σ,i + |Dσ| (∇p)n+1σ,i un+1

σ,i = 0. (3.46)


Dividing the velocity correction equation (3.42c) by (|Dσ|δt

ρnDσ

)12 , we obtain:

[|Dσ|δt

ρnDσ

]1/2

un+1σ,i +

[|Dσ| δtρn

Dσ

]1/2

(∇p)n+1σ,i =

[|Dσ|δt

ρnDσ

]1/2

un+1σ,i +

[|Dσ| δtρn

Dσ

]1/2

(∇p)n+1σ,i .

Squaring this relation, dividing by two and summing it with (3.46) yields the result, using the definition (3.7)of (ΔMu)n+1. �

The discrete potential balance is again valid for the pressure correction algorithm, thanks to the fact thatthe mass balance (3.42d) is satisfied. The proof is identical to that of Lemma 3.2 given for the implicit scheme.

Lemma 3.12 (Discrete potential balance, pressure correction scheme). A solution to the system (3.42) satisfiesthe discrete potential balance (3.19), with Rn+1

K defined by (3.20).

Let us now turn to the entropy inequality.

Proposition 3.13 (Global discrete entropy inequality, existence of a solution). Assume that the initial conditionρ0 is positive. Then there exists a solution (un, ρn) 0≤n≤N and (un) 1≤n≤N to the scheme (3.42), and, for1 ≤ n ≤ N , ρn > 0 and the following inequality holds:

12

d∑i=1

∑σ∈E(i)

S

|Dσ| ρn−1Dσ

(unσ,i)

2 +∑

K∈M|K| H(ρn

K) + Rn ≤ C0, (3.47)

where C0 ∈ R+ only depends on the initial conditions and on the density field ρ0 computed at the initializationof the algorithm. The remainder term Rn is non-negative, and gathers some estimates of the space and timetranslates of the unknowns:

Rn =d∑

i=1

n∑k=1

⎡⎢⎣12

∑σ∈E(i)

S

|Dσ| ρk−2Dσ

(ukσ,i − uk−1

σ,i )2 + δt∑

ε=Dσ |Dσ′∈E(i)S

ν hd−2σ (uk

σ,i − ukσ′,i)

2

⎤⎥⎦+γ

2

n∑k=1

δt∑

σ=K|L∈Eint

|σ| (ρkσ,γ)γ−2 |uk

K,σ| (ρkK − ρk

L)2 +δt2

2

∑σ∈Eint

|Dσ|ρn−1

Dσ

|(∇p)nσ |2, (3.48)

with ρkσ,γ equal to either ρk

K or ρkL and such that (ρk

σ,γ)γ−2 = min((ρk

K)γ−2, (ρkL)γ−2

).

Proof. The essential arguments of the proof of this proposition are given in ([14], Thm. 3.8) with slightly differentnotations, so we briefly recall here how to obtain the estimate (3.47), for the sake of completeness. We sum thekinetic energy balance equation (3.44) over the faces, and the elastic potential balance (3.19) over the cells, andfinally sum the two obtained relations. We obtain a “local in time” version of equation (3.47), which reads:

T n+1 − T n + Rn+1 + Pn+1 = 0, (3.49)

where:

T n+1 =∑

K∈M|K| H(ρn+1

K ) +12

d∑i=1

∑σ∈E(i)

S

|Dσ| ρnDσ

(un+1σ,i )2,

and:Rn+1 =

∑1≤i≤d

∑σ∈E(i)

S

Rn+1σ,i , Pn+1 =

∑1≤i≤d

∑σ∈E(i)

S ∩Eint

Pn+1σ,i ,


with Rn+1σ,i , and Pn+1

σ,i given by equation (3.45). The term Pn+1 thus reads:

Pn+1 =δt2

2

∑σ∈Eint

|Dσ|ρn

Dσ

[|(∇p)n+1

σ |2 − |(∇p)n+1σ |2

]=δt2

2

∑σ∈Eint

|Dσ|[|(∇p)n+1

σ |2ρn

Dσ

− |(∇p)nσ|2

ρn−1Dσ

]· (3.50)

Then, summing (3.49) over the time steps yields the estimate (3.47) with:

C0 =∑

K∈M|K| H(ρ0

K) +12

∑1≤i≤d

∑σ∈E(i)

S

|Dσ| ρ−1Dσ

(u0σ,i)

2 +δt2

2

∑σ∈Eint

|Dσ|ρ−1

Dσ

|(∇p)0σ|2. �

Remark 3.14. As in the implicit case, the inequality (3.47) thus provides an estimate on the unknowns (seeRem. 3.4).

Remark 3.15 (Regularity assumptions for the initial conditions). For a given mesh, the quantity denotedabove by C0 is bounded whenever ρ0 is positive and belongs to L1(Ω) and u0 belongs to L1(Ω)d. When dealingwith a sequence of discretizations to pass to the limit in the scheme, we need to assume that C0 is controlledindependently of the mesh and time step, which necessitates (i) that the initial kinetic energy is bounded, (ii)that H(ρ0

K) is bounded in L1(Ω), and (iii) that the last term involving the discrete pressure gradient does notblow-up.

Assumption (ii) (and, of course, (i)) may be obtained by supposing that both u0 and ρ0 belongs to L∞(Ω)and L∞(Ω)d respectively and that δt/h is bounded (possibly by a number much larger than 1); indeed, ρ0 is thenobtained in this case by a single time step of a (discrete) transport equation with a velocity field the divergenceof which is controlled by 1/h, and so ρ0 is controlled in L∞(Ω). Assumption (iii) may then be inferred by thesame assumption on the ratio δt/h, together with the hypothesis that the data ρ0 (and so ρ−1) is bounded awayfrom zero. Indeed, since ρ0 is bounded, so is p0 and we get:

∑σ∈Eint

|Dσ| δt2

ρ−1Dσ

|(∇p)0σ|2 =∑

σ=K|L∈Eint

δt2 |σ|2

ρ−1Dσ

|Dσ|(p0

K − p0L)2 ≤ C ‖ 1

ρ−1‖L∞(Ω)

‖p0‖2

L∞(Ω)

∑σ∈Eint

h2 |σ|2|Dσ|

and the last sum is bounded. We shall work under these assumptions for the passage to the limit in the scheme.

The expression (3.50) in the proof of Proposition 3.13, where Pn+1 is set under the form of a differenceof the same two quantities taken at two consecutive time steps, shows the interest of the pressure gradientrenormalization step. We prove in addition in the following lemma that, under stability conditions, this remainderterm tends to zero in a discrete distribution sense.

Lemma 3.16 (Pressure remainder terms). Let (M(m), δt(m))m∈N be a sequence of meshes and time steps, suchthat h(m) and δt(m) tend to zero as m tends to infinity, and satisfying the CFL-like condition:

∀m ∈ N,δt(m)

h(m)≤ C, with h(m) = min

σ∈E(m)int

|Dσ||σ| ,

and where C is a positive real number which can be greater than 1. Let (ρ(m))m∈N and (p(m))m∈N be (part of) theassociated sequence of discrete solutions, satisfying equations (3.42). We assume that the sequence (p(m))m∈N

is uniformly bounded in L∞(Ω × (0, T )) and in the discrete L1(0, T ; BV (Ω)) norm:

∀m ∈ N, ‖p(m)‖T ,x,BV =N(m)∑n=0

δt∑

σ=K|L∈E(m)int

|σ|∣∣(p(m))n

L − (p(m))nK

∣∣ ≤ C. (3.51)


We furthermore suppose that (ρ(m))m∈N and (1/ρ(m))m∈N are bounded in L∞(Ω×(0, T )). Let ϕ ∈ C∞c (Ω×[0, T )),

and, for m ∈ N, 0 ≤ n ≤ N (m) and σ ∈ E(m)int , let ϕn

σ = ϕ(xσ, tn), with xσ the mass center of σ. Let us define

the quantity Pn+1σ , for 0 ≤ n ≤ N − 1 and σ ∈ Eint, as:

Pn+1σ =

|Dσ| δtρn

Dσ

[|(∇p)n+1

σ |2 − |(∇p)n+1σ |2

]. (3.52)

Then

limm→+∞

[n−1∑n=0

δt∑

σ∈Eint

Pn+1σ ϕn

σ

]= 0.

Proof. By definition of Pn+1σ , we get that

N−1∑n=0

δt∑

σ=K|L∈Eint

Pn+1σ ϕn

σ =N−1∑n=0

δt2∑

σ=K|L∈Eint

|σ|2|Dσ|

[1ρn

Dσ

(pn+1K − pn+1

L )2 − 1ρn−1

Dσ

(pnK − pn

L)2]ϕn

σ .

A discrete integration by parts yields:∣∣∣∣∣∣N−1∑n=0

δt∑

σ=K|L∈Eint

Pn+1σ ϕn

σ

∣∣∣∣∣∣ ≤ δt2


1ρ−1

Dσ

|σ|2|Dσ|

|p0K − p0

L|2 |ϕ0σ |

+N−1∑n=0

∑σ=K|L∈Eint

1ρn

Dσ

|σ|2|Dσ|

(pn+1K − pn+1

L )2|ϕn+1σ − ϕn

σ|

⎤⎦ .Using the fact that, for 0 ≤ n ≤ N and σ = K|L ∈ Eint, (pn

K − pnL)2 ≤ 2 ‖p‖L∞(Ω×(0,T )) |pn

K − pnL|, we get,

thanks to the regularity of ϕ, that

|N−1∑n=0

δt∑

σ=K|L∈Eint

Pn+1σ ϕn

σ| ≤ Cϕδt2

h‖p‖L∞(Ω×(0,T ))

[‖p0‖BV (Ω) + ‖p‖T ,x,BV

],

which concludes the proof. �

3.2.3. Passing to the limit in the scheme

Using the notations of Section 3.1.3, the pressure correction scheme in one space dimension reads:

Initialization – Compute ρ−1, u0, solve for ρ0 and compute p0:

∀K ∈ M, ρ−1K =

1|K|

∫K

ρ0(x) dx,

∀σ ∈ Eint, u0σ =

1|Dσ|

∫Dσ

u0(x) dx,

∀K = [−→σσ′] ∈ M,

|K|δt

(ρ0K − ρ−1

K ) + F 0σ′ − F 0

σ = 0,

∀K ∈ M, p0K = (ρ0

K)γ .

(3.53a)


∀σ =−−→K|L ∈ Eint, (δp)n+1

σ =

√ρn

Dσ

ρn−1Dσ

(pnL − pn

K). (3.53b)



∀σ =−−→K|L ∈ Eint,

|Dσ|δt

(ρnDσun+1

σ − ρn−1Dσ

unσ) + Fn

L un+1L − Fn

K un+1K

−|Dσ| (ΔMu)n+1σ + (δp)n+1

σ = 0,(3.53c)

Correction step – Solve for ρn+1, pn+1 and un+1:

∀σ =−−→K|L ∈ Eint,

|Dσ|δt

ρnDσ

(un+1σ − un+1

σ ) + (pn+1L − pn+1

K ) − (δp)n+1σ = 0, (3.53d)

∀K = [−→σσ′] ∈ M,

|K|δt

(ρn+1K − ρn

K) + Fn+1σ′ − Fn+1

σ = 0, (3.53e)

∀K ∈ M, pn+1K = (ρn+1

K )γ . (3.53f)

Theorem 3.17 (Consistency of the pressure correction scheme). Let Ω be an open bounded interval of R. Wesuppose that ρ0 ∈ L∞(Ω), 1/ρ0 ∈ L∞(Ω) and u0 ∈ L∞(Ω). Let (M(m), δt(m), ν(m))m∈N be a regular sequence ofdiscretizations in the sense of Definition 3.5, and let (ρ(m), p(m), u(m), u(m))m∈N be the corresponding sequenceof solutions. We suppose that this sequence converges in Lp(Ω × (0, T ))4, for 1 ≤ p < +∞, to (ρ, p, u, ¯u) ∈L∞(Ω×(0, T ))4. We suppose in addition that both sequences (ρ(m))m∈N and (1/ρ(m))m∈N are uniformly boundedin L∞(Ω × (0, T )).

Then u = ¯u and the triplet (ρ, p, u) satisfies the system (3.28).

Proof. Let m ∈ N be given. Dropping for short the superscript (m), the estimate of Proposition 3.13 yields:n∑

k=1

δt∑

σ∈Eint

|Dσ| ρk−1Dσ

(ukσ − uk−1

σ )2 ≤ C δt, (3.54)

where, by the assumption on the initial data, the real number C is independent of m (see Rem. 3.15). Hence,

‖u(m) − u(m)(., .− δt)‖2

L2(Ω×(0,T )) ≤ C δt(m) ‖ 1ρ(m)

‖L∞(Ω×(0,T ))

.

Letting m tend to +∞ in this equation yields u = ¯u.As for the implicit scheme, with the assumed convergence for the sequence of solutions, the limit satisfies

the equation of state. The passage to the limit in the mass balance equation is the same as in the implicitcase, and we only need to address here the momentum balance equation. Let ϕ ∈ C∞

c (Ω × [0, T )), and let itsinterpolate ϕE and its discrete derivatives be defined by Definition 3.6. Summing the velocity prediction andcorrection equations, multiplying the result by δt ϕn

σ and then summing over the faces and time steps, we getT

(m)1 + T

(m)2 + T

(m)3 + T

(m)4 = 0, with:

T(m)1 =

N−1∑n=0

∑σ∈Eint

|Dσ|[ρn

Dσun+1

σ − ρn−1Dσ

unσ

]ϕn

σ ,

T(m)2 =

N−1∑n=0

δt∑


[Fn

L un+1L − Fn

K un+1K

]ϕn

σ,


T(m)3 =

N−1∑n=0

δt∑


(pn+1L − pn+1

K ) ϕnσ ,

T(m)4 =

N−1∑n=0

δt∑

σ∈Eint

[ ∑K=[σσ′ ]

ν

hK(un+1

σ − un+1σ′ )]ϕn

σ .

Thanks to the definition (3.26) of the density on the dual mesh ρDσ , reordering the sums, we get for T (m)1 :

T(m)1 = −

N−1∑n=0

δt∑

σ=K|L∈Eint

[ |K|2ρn

K +|L|2ρn

L

]un+1

σ

ϕn+1σ − ϕn

σ

δt−

∑σ=K|L∈Eint

[ |K|2ρ−1

K +|L|2ρ−1

L

]u0

σ ϕ0σ

= −∫ T

0

∫Ω

ρ(m) u(m)ðt ϕM(m) dxdt−

∫Ω

(ρ(m))−1(x) (u(m))0(x) ϕM(m)(x, 0) dx.

From the definition (3.53a) of the initial conditions, the sequences((ρ(m))−1

)and(u(m))0

)converge in Lr(Ω),

for r ≥ 1, to ρ0 and u0 respectively. Thanks to the convergence assumption for the sequence of discrete solutions,and noting that the sequence

(ρ(m)(·, · − δt)

)m∈N

converges to ρ as (ρ(m))m∈N, we get:

limm−→+∞

T(m)1 = −

∫ T

0

∫Ω

ρ u ∂tϕdxdt −∫

Ω

ρ0(x) u0(x) ϕ(x, 0) dx.

Let us now turn to T (m)2 . From the expression (3.26) of the fluxes FK and the values uK , reordering the sums,

we get:

T(m)2 = −1

4

N−1∑n=0

δt∑

K=[−−→σσ′]∈M

(ρnσu

nσ + ρn

σ′ unσ′) (un+1

σ + un+1σ′ ) (ϕn

σ′ − ϕnσ),

which we write T (m)2 = T (m)

2 + R(m)2 with:

T (m)2 = −1

2

N−1∑n=0

δt∑

K=[−−→σσ′]∈M

ρnK

[un

σ un+1σ + un

σ′ un+1σ′

](ϕn

σ′ − ϕnσ). (3.56)

= −∫ T

0

∫Ω

ρ(m)(·, · − δt) u(m)(·, · − δt) u(m)ðxϕE dxdt, (3.57)

and therefore,

limm−→+∞

T (m)2 = −

∫ T

0

∫Ω

ρ u2 ∂xϕdxdt.


R(m)2 = − 1

4

N−1∑n=0

δt∑

K=[−−→σσ′]∈M

[(ρn

σunσ + ρn

σ′unσ′)(un+1

σ + un+1σ′)

− 2ρnK

(un

σ un+1σ + un

σ′ un+1σ′)]

(ϕnσ′ − ϕn

σ) .


Expanding the quantity 2ρnK (un

σ un+1σ +un

σ′ un+1σ′ ) thanks to the identity 2(ab+cd) = (a+c)(b+d)+(a−c)(b−d),

we get R(m)2 = R(m)

2,1 + R(m)2,2 :

R(m)2,1 = −1

4

N−1∑n=0

δt∑

K=[−−→σσ′ ]∈M

((ρn

σ − ρnK) un

σ + (ρnσ′ − ρn

K) unσ′

)(un+1

σ + un+1σ′ ) (ϕn

σ′ − ϕnσ),

R(m)2,2 =

14

N−1∑n=0

δt∑

K=[−−→σσ′]∈M

ρnK (un

σ − unσ′) (un+1

σ − un+1σ′ ) (ϕn

σ′ − ϕnσ).

First we study R(m)2,1 . Thanks to the definition of the upwind approximation, reordering the sum by faces, we

get:

R(m)2,1 =

14

N−1∑n=0

δt∑

σ∈Eint,


εnσ (ρn

L − ρnK) un

σ (un+1σ + un+1

σ′ ) (ϕnσ − ϕn

σ′ ),

where we recall that the notation σ = L→ K means that the face σ separates K and L and the flow goes fromL to K at the time level n, and where εn

σ = ±1. Since |ϕnσ − ϕn

σ′ | ≤ Cϕ |K| ≤ Cϕ (|Dσ| + |Dσ′ |), we get:

|R(m)2,1 | ≤ Cϕ

4

N−1∑n=0

δt∑

σ∈Eint,


(|Dσ| + |Dσ′ |

)|ρn

L − ρnK | |un

σ| |un+1σ + un+1

σ′ |.

Therefore, by the Cauchy−Schwarz inequality, we get:

|R(m)2,1 | ≤ Cϕ

4(h(m))1/2

[N−1∑n=0

δt∑

σ=K|L∈Eint

|unσ| (ρn

L − ρnK)2]1/2

×[N−1∑

n=0

δt∑

σ∈Eint,


(|Dσ| + |Dσ′ |

)|un

σ|(un+1

σ + un+1σ′)2]1/2

.

Since the ratio of the size of two neighboring meshes is bounded by the regularity assumption on the mesh, weget from the estimate (3.47) on the solution:

|R(m)2,1 | ≤ C (h(m))1/2

[‖u(m)‖L2(Ω×(0,T )) + ‖u(m)‖2

L4(Ω×(0,T ))

], (3.58)

where C ∈ R+ does not depend on m, and so R(m)2,1 tends to zero when m tends to +∞. For R(m)

2,2 , we have, bythe Cauchy−Schwarz inequality:

|R(m)2,2 | ≤ Cϕ

N−1∑n=0

δt∑

K=[−−→σσ′]∈M

|K| ρn+1K |un

σ + unσ′ | (un+1

σ − un+1σ′ )

≤ Cϕh(m)

(ν(m))1/2‖ρ(m)‖L∞(Ω×(0,T )) ‖u(m)‖L2(Ω×(0,T ))

⎡⎣N−1∑n=0

δt∑

K=[σσ′]∈M

ν(m)

hK(un+1

σ − un+1σ′ )2

⎤⎦1/2

,

and thus, thanks to the estimate (3.47):

|R(m)2,2 | ≤ C

h(m)

(ν(m))1/2‖ρ(m)‖L∞(Ω×(0,T )) ‖u(m)‖L2(Ω×(0,T )),


where C ∈ R+ does not depend on m. Therefore, this term also tends to zero when m tends to +∞.Next, we turn to the term T

(m)3 :

T(m)3 = −

N−1∑n=0

δt∑

K=[−−→σσ′]∈M

|K| pn+1K

ϕnσ′ − ϕn

σ

hK= −

∫ T

0

∫Ω

p(m)ðxϕE dxdt,

and therefore,

limm−→+∞

T(m)3 = −

∫ T

0

∫Ω

p ∂xϕdxdt.

Let us finally study T (m)4 . Reordering the sums, we get:

T(m)4 =

N−1∑n=0

δt∑

K=[−−→σσ′]∈M

ν(m)

hK(un+1

σ − un+1σ′ ) (ϕn

σ − ϕnσ′ ).

The Cauchy−Schwarz inequality yields:

|T (m)4 | ≤

⎡⎢⎣N−1∑n=0

δt∑

K=[−−→σσ′]∈M

ν(m)

hK(un+1

σ − un+1σ′ )2

⎤⎥⎦1/2 ⎡⎢⎣N−1∑

n=0

δt∑

K=[−−→σσ′]∈M

ν(m)

hK(ϕn

σ − ϕnσ′ )2

⎤⎥⎦1/2

,

and thus, again in view of the estimate (3.47), this term tends to zero when ν(m) tends to zero. �

Remark 3.18 (On the “non appearance of void assumption”). The assumption that (1/ρ(m))m∈N is boundedin L∞(Ω× (0, T )) (which, loosely speaking, means that the appearance of void is excluded) is used twice in theproof of Theorem 3.17. We use it for the first time to obtain u = ¯u. Here, the hypothesis may be circumventedby replacing this conclusion by ρu = ρ¯u (or, in other words, u = ¯u wherever ρ = 0), which is easily obtainedfrom inequality (3.54) below. The second time is to obtain, as in the implicit case, the “non–weighted” estimateof the density space translates (3.32) for γ ≥ 2, see Remark 3.8.

Remark 3.19 (Less sharp bounds and more general meshes). As in the implicit case, the assumption that theratio of the size of two neighboring cells is bounded, i.e. Assumption (ii) of Definition 3.5, is only used for theremainder associated with the the convection term in the momentum balance. It may be avoided if we supposethat the sequence of solution is uniformly bounded, replacing (3.58) by

|R(m)2,1 | ≤ C (h(m))1/2 ‖u(m)‖1/2

L∞(Ω×(0,T )) ‖u(m)‖L∞(Ω×(0,T )).

We now turn to the entropy condition. For any piecewise constant function q on primal cells, we define itsL1(0, T ; BV (Ω)) norm by:

‖q‖T ,x,BV =N∑

n=0

δt∑

σ=K|L∈Eint

|qnL − qn

K |. (3.59)

With this notation, we are now in position to state the following result.


Theorem 3.20 (Entropy consistency, pressure correction scheme). Under the assumptions of Theorem 3.17,we furthermore assume that:

- the sequence of regular meshes satisfies:

∀m ∈ N,δt(m)


σ∈E(m)int

|hσ|,

and where C is a positive real number which can be greater than 1,- the sequence (p(m))m∈N is uniformly bounded in the discrete L1(0, T ; BV (Ω)) norm defined by (3.59).

Then the limit (ρ, p, u) satisfies the entropy condition (3.35).

Proof. Let ϕ ∈ C∞c

(Ω × [0, T )

), ϕ ≥ 0. Again using the notations of Definition 3.6, we multiply the kinetic

balance equation (3.44) by ϕnσ, and the elastic potential balance (3.19) by ϕn

K , sum over the faces and cellsrespectively and over the time steps, to get:∑

σ∈Eint

T n+1σ ϕn

σ +∑

K∈MT n+1

K ϕnK = −

∑σ∈Eint

Rn+1σ ϕn

σ −∑

K∈MRn+1

K ϕnK −

∑σ∈Eint

Pn+1σ ϕn

σ, (3.60)

where, for σ =−−→K|L, K = [

−→σ′σ] and L = [

−−→σσ′′],

T n+1σ =

12|Dσ|δt

[ρn

Dσ(un+1

σ )2 − ρn−1Dσ

(unσ)2]

+12Fn+1

L un+1σ un+1

σ′′ − 12Fn+1

K un+1σ un+1

σ′ + (pn+1L − pn+1

K ) un+1σ ,

for K = [−→σσ′],

T n+1K =

|K|δt

[H(ρn+1

K ) −H(ρnK)]

+ un+1σ′ H(ρn+1

σ′ ) − un+1σ H(ρn+1

σ ) + pn+1K (un+1

σ′ − un+1σ ),

the quantities Rn+1σ and Pn+1

σ are given by (the one-dimensional version of) equation (3.45), and Rn+1K is given

by (the one-dimensional version of) equation (3.20).The discrete weak form of the entropy balance is obtained by integrating in time (i.e. summing over the

time steps) equation (3.60). We obtain T(m)1 + T

(m)2 + T

(m)3 + T

(m)4 + T

(m)5 + R(m) + P (m) = 0 where T (m)

1 ,T

(m)2 , T (m)

4 , T (m)5 and R(m) are identical to their namesakes in the implicit case (see proof of Thm. 3.10, defined

by (3.37a), (3.37b), (3.37d), (3.37e) and (3.37f) respectively, and

T(m)3 =

12

N−1∑n=0

δt∑


K=[−−→σ′σ], L=[

−−→σσ′′]

[Fn+1

L un+1σ un+1

σ′′ − Fn+1K un+1

σ un+1σ′

]ϕn

σ, (3.61a)

P (m) =N−1∑n=0

δt∑

σ∈Eint

Pn+1σ ϕn

σ. (3.61b)

The terms T (m)1 , T (m)

2 , T (m)4 , T (m)

5 and R(m) were studied in the proof of Theorem 3.10. Let us then study thekinetic energy convection term T

(m)3 which reads, after reordering the summations:

T(m)3 = −1

2

N−1∑n=0

δt∑

K=[−−→σσ′]∈M

FnK un+1

σ un+1σ′ (ϕn

σ′ − ϕnσ).


We write Tm3 = T (m)

3 + R(m)3 , where

T (m)3 = −1

2

N−1∑n=0

δt∑

K=[−−→σσ′]∈M

|K|2

ρnK

[un

σ(un+1σ )2 + un

σ′(un+1σ′ )2

] ϕnσ′ − ϕn

σ

hK

= −12

∫ T

0

∫Ω

ρ(m)(x, t− δt) u(m)(x, t− δt) (u(m)(x, t))2 ðxϕE dxdt,

so that

limm−→+∞

T (m)3 = −1

2

∫ T

0

∫Ω

ρ u3 ∂xϕdxdt.


R(m)3 = −1

4

N−1∑n=0

δt∑

K=[−−→σσ′]∈M

[(ρn

σunσ + ρn

σ′unσ′) un+1

σ′ un+1σ − ρn

K

(un

σ (un+1σ )2 + un

σ′ (un+1σ′ )2

)](ϕn

σ′ − ϕnσ).

Reordering the terms in the sum, we get:

R(m)3 = −1

4

N−1∑n=0

δt∑

K=[−−→σσ′]∈M

⎡⎢⎢⎣(ρnσ u

n+1σ′ − ρn

K un+1σ )un

σ un+1σ︸︷︷︸

D1

+ (ρnσ′ un+1

σ − ρnK u

n+1σ′ )un

σ′ un+1σ′︸︷︷︸

D2

⎤⎥⎥⎦ (ϕnσ′ − ϕn

σ).

Let us consider the term involving D1, and skip the exposition of the treatment of the term with D2, whichis similar. Using the identity 2(ab − cd) = (a − c)(b + d) + (a + c)(b − d), we split this first part of R(m)

3 intoR(m)

31 + R(m)32 , with:

R(m)31 = −1

8

N−1∑n=0

δt∑

K=[−−→σσ′]∈M

unσ u

n+1σ (ρn

σ − ρnK) (un+1

σ′ + un+1σ ) (ϕn

σ′ − ϕnσ),

R(m)32 = −1

8

N−1∑n=0

δt∑

K=[−−→σσ′]∈M

unσ u

n+1σ (ρn

σ + ρnK) (un+1

σ′ − un+1σ ) (ϕn

σ′ − ϕnσ).

Thanks to the regularity of ϕ, the Cauchy−Schwarz inequality yields:

|R(m)31 | ≤ Cϕ h1/2

⎡⎢⎣N−1∑n=0

δt∑

K=[−−→σσ′ ]∈M

|unσ| (ρn

σ − ρnK)2

⎤⎥⎦1/2⎡⎢⎣N−1∑

n=0

δt∑

K=[−−→σσ′]∈M

|K| |unσ|(un+1

σ (un+1σ′ + un+1

σ ))2⎤⎥⎦

1/2

,

and thus, invoking the estimate (3.47),

|R(m)31 | ≤ C (h(m))1/2

[‖u(m)‖L2(Ω×(0,T )) + ‖u(m)‖4

L8(Ω×(0,T ))

].

Similarly, we get:

|R(m)32 | ≤ Cϕ

h

ν1/2

⎡⎢⎣N−1∑n=0

δt∑

K=[−−→σσ′]∈M

ν

hK(un+1

σ′ − un+1σ )2

⎤⎥⎦1/2 ⎡⎢⎣N−1∑

n=0

δt∑

K=[−−→σσ′]∈M

|K|(un

σun+1σ (ρn

σ + ρnK))2⎤⎥⎦

1/2

,


and thus:

|R(m)32 | ≤ C

h(m)

(ν(m))1/2

[‖u(m)‖3

L6(Ω×(0,T )) + ‖u(m)‖3

L6(Ω×(0,T )) +[‖ρ(m)‖3

L6(Ω×(0,T ))

].

Therefore, under the assumptions of the theorem, we have

limm−→+∞

T(m)3 = −1

2

∫ T

0

∫Ω

ρ u3 ∂xϕdxdt. (3.62)

Together with the results which were obtained in the proof of Theorem 3.10, this yields

limm−→+∞

( 5∑i=1

T(m)i +R(m)) ≥ −

∫ T

0

∫Ω

[η ∂tϕ+ (η + p)u ∂xϕ] dxdt−∫

Ω

η0(x) ϕ(x, 0) dx.

In the proof of Theorem 3.10, we obtained that limm→+∞R(m) ≥ 0. Furthermore, Lemma 3.16 implies thatlimm→+∞ P (m) = 0, which concludes the proof of the theorem. �

4. The pressure correction scheme for the full Euler equations

We now turn to the development and study of a similar pressure correction scheme for the full Euler equa-tions, that is for the system (1.2) assuming τ (u) = 0. Numerical schemes for the Euler equations have beenwidely studied, and are very often based on Riemann solvers on the system consisting of the mass balance, themomentum balance, and the total energy balance. We start this section by explaining why we use the internalenergy balance rather the the total energy in the correction scheme, and the precautions that must be taken inorder to compute correct shocks in this way.

4.1. Internal energy, kinetic energy and total energy

Let us suppose that the solution to the Navier–Stokes equations (1.2) is regular. As already mentioned, takingthe inner product of the momentum balance equation (1.2b) by u and using the mass balance equation, weobtain the so-called kinetic energy balance equation:

12∂t(ρ |u|2) +

12div(ρ |u|2u) + ∇p · u = div(τ (u)) · u. (4.1)

Subtracting this relation from the total energy balance, we obtain the internal energy balance equation:

∂t(ρe) + div(ρeu) + p div(u) = τ (u) : ∇u. (4.2)

Since,

– from thermodynamical arguments, τ (u) : ∇u ≥ 0;– thanks to the mass balance equation, the first two terms in the left-hand side of (4.2) may be recast as a

transport operator: ∂t(ρe) + div(ρeu) = ρ [∂te+ u · ∇e];– and, finally, from the equation of state, the pressure vanishes when e = 0,

this equation implies that, if e ≥ 0 at t = 0 and with suitable boundary conditions, then e remains non-negativeat all times.

Our aim here is to build a scheme stable and accurate at all Mach numbers, and, in particular, which boilsdown to a usual scheme for incompressible flows (or, more generally speaking, for the asymptotic model ofvanishing Mach number flows [45]) when the Mach number tends to zero. In these latter models, the naturalenergy balance equation is the internal energy equation (4.2). In addition, discretizing (4.2) instead of the totalenergy balance (1.2c) presents two advantages:


– first, it avoids the space discretization of the total energy, which is rather unnatural for staggered schemessince the degrees of freedom for the velocity and the scalar variables are not colocated,

– second, a suitable discretization of (4.2) may yield, “by construction” of the scheme, the positivity of theinternal energy [18].

However, in the inviscid case and for solutions with shocks, equation (4.2) (with τ = 0) is not equivalentto the conservative total energy balance (1.2c) (with τ = 0); more precisely speaking, at the locations ofshocks, positive measures should replace, at the right-hand side of equation (4.2), the term τ (u) : ∇u whichis formally the product of vanishing quantities (for a Newtonian fluid, the viscosity) and infinite derivatives ofthe velocity. Discretizing (4.2) instead of (1.2c) may thus yield a scheme which does not compute the correctweak discontinuous solutions; in particular, the numerical solutions may present (smeared) shocks which donot satisfy the Rankine–Hugoniot conditions associated with (1.2c). The essential result of this section is toprovide a solution to circumvent this problem. To this purpose, we closely mimic the above performed formalcomputation:

– Starting from the discrete momentum balance equation, we derive a discrete kinetic energy balance (in fact,this computation is already performed in the previous sections, since we use for the full Euler equations thesame discrete momentum balance as in the barotropic case). In this relation, residual terms which do notend to zero with space and time step appear (they are the discrete manifestations of the above mentionedmeasures).

– These residual terms are then compensated by corrective terms in the internal energy balance.

We provide a theoretical justification of this process by showing that, in the 1D case, if the scheme is stableand converges to a limit (in a sense to be defined), this limit satisfies a weak form of (1.2c) which implies thecorrect Rankine–Hugoniot conditions. Then, we perform numerical tests which substantiate this analysis. Afully implicit scheme was studied in [26] along with two pressure correction schemes: the first correction schemeis appealing for its (relative) simplicity, but does not seem to warrant the sign of the internal energy (so thatthe unconditional stability induced by the above mentioned conservation of the total energy property is lost);the second scheme cures this problem, at the price of the introduction of an additional elliptic problem whichmust be solved at the beginning of each time step to determine a tentative pressure. Here, we present a variantof these schemes, that preserves the positivity of the internal energy thanks to a renormalization step whichonly consists in a weighting of the discrete pressure gradient and does not require any elliptic solve (and whichis thus much less costly). As the correction scheme for barotropic flows, it is implemented in the industrialopen-source code CALIF3S [5]. Let us mention also that fully explicit versions have been studied [30, 31].

4.2. The scheme

We propose in this section a pressure correction scheme, which, as in the barotropic case, features a renor-malization step for the pressure gradient. As previously mentioned, we add a corrective term in the internalenergy equation; we are able to show that this corrective term is non negative, which ensures the positivity ofthe internal energy and the existence of a solution to the scheme.

With the notations that were introduced in Section 3.2.1, the algorithm reads, for 0 ≤ n ≤ N − 1:


∀σ ∈ E , (∇p)n+1σ =

√ρn

Dσ

ρn−1Dσ

(∇p)nσ. (4.3a)



For 1 ≤ i ≤ d, ∀σ ∈ E(i)S ,

|Dσ|δt

(ρnDσun+1

σ,i − ρn−1Dσ

unσ,i) +

∑ε∈E(Dσ)

Fnσ,εu

n+1ε,i − |Dσ| (ΔMu)n+1

σ,i + |Dσ| (∇p)n+1σ,i = 0. (4.3b)

Correction step – Solve for ρn+1, pn+1, en+1 and un+1:

For 1 ≤ i ≤ d, ∀σ ∈ E(i)S ,

|Dσ|δt

ρnDσ

(un+1σ,i − un+1

σ,i ) + |Dσ|[(∇p)n+1

σ,i − (∇p)n+1σ,i

]= 0, (4.3c)

∀K ∈ M,|K|δt

(ρn+1K − ρn

K) +∑

σ∈E(K)

Fn+1K,σ = 0, (4.3d)

∀K ∈ M,|K|δt

(ρn+1K en+1

K − ρnKe

nK) +

∑σ∈E(K)

Fn+1K,σ e

n+1σ

+|K| pn+1K (divu)n+1

K = Sn+1K ,

(4.3e)

∀K ∈ M, pn+1K = (γ − 1) ρn+1

K en+1K , (4.3f)

where we make an upwind choice for e (again a crucial choice to ensure the positivity of the internal energy):

for σ = K|L ∈ Eint, en+1σ =

∣∣∣∣∣ en+1K if Fn+1

K,σ ≥ 0,

en+1L otherwise.

(4.4)

The initialization of the scheme is performed in a way similar to the barotropic case. First, ρ−1, e0 and u0 aregiven by the average of the initial conditions ρ0, e0 and u0, i.e. by (3.43) and the following relation:

∀K ∈ M, e0K =1|K|

∫K

e0(x) dx. (4.5)

Then, we compute ρ0 by solving the mass balance equation (4.3d). Finally, the initial pressure p0 is computedfrom the initial density ρ0 by the equation of state: ∀K ∈ M, p0

K = (γ − 1) ρ0K e0K . As in the barotropic case,

the objective of this procedure is to perform the first prediction step with (ρ−1Dσ

)σ∈E , (ρ0Dσ

)σ∈E and the dualmass fluxes satisfying the mass balance.

There only remains to define the corrective terms Sn+1K in the internal energy balance (4.3e), with the aim

to recover a consistent discretization of the total energy balance. The first idea to do this could be just tosum the (discrete) kinetic energy balance with the internal energy balance: it is indeed possible for a colocateddiscretization. But here, we face the fact that the kinetic energy balance is associated with the dual mesh, whilethe internal energy balance is discretized on the primal one. The way to circumvent this difficulty is to remarkthat we do not really need a discrete total energy balance; in fact, we only need to recover (a weak form of)this equation when the mesh and time steps tend to zero. To this purpose, we choose the quantities (Sn+1

K ) insuch a way as to somewhat compensate the terms (Rn+1

σ,i ) defined by (3.45) appearing at the right-hand-side ofthe discrete kinetic energy identity (3.44):

∀K ∈ M, Sn+1K =

d∑i=1

Sn+1K,i ,


with:

Sn+1K,i =

12ρn−1

K

∑σ∈E(K)∩E(i)

S

|DK,σ|δt

(un+1

σ,i − unσ,i

)2+

∑ε∈E(i)

S , ε∩K =∅,ε=Dσ|Dσ′

αK,ε ν hd−2ε

(un+1


)2

. (4.6)

The coefficient αK,ε is fixed to 1 if the face ε is included in K, and this is the only situation to consider for theRT and CR discretizations. For the MAC scheme, some dual faces are included in the primal cells, but some lieon their boundary; for such a dual face ε, we denote by Nε the set of cells M such that M ∩ ε = ∅ (the cardinalof this set being always 4), and compute αK,ε by:

αK,ε =|K|∑

M∈Nε|M | ·

For a uniform grid, this formula yields αK,ε = 1/4.The expression of the terms(Sn+1

K )K∈M is justified by the passage to the limit in the scheme (for a one-dimensional problem) performed in Section 4.3. However, its expression may be anticipated, thanks to thefollowing remarks. First, we note that:

∑K∈M

Sn+1K −

d∑i=1

∑σ∈E(i)

S

Rn+1σ,i = 0. (4.7)

Indeed, for K ∈ M and σ = K|L, the first part of Sn+1K,i , thanks to the expression (3.4) of the density at the

face ρnDσ

, results from a dispatching of the first part of the kinetic energy balance residual Rn+1σ,i over the two

cells adjacent to σ:

12

|Dσ|δt

ρn−1Dσ

(un+1

σ,i − unσ,i

)2 =12

|DK,σ|δt

ρn−1K

(un+1

σ,i − unσ,i

)2︸︷︷︸affected to K

+12

|DL,σ|δt

ρn−1L

(un+1

σ,i − unσ,i

)2︸︷︷︸affected to L

.

For the second part of the remainder, a standard reordering of the sum yields:

d∑i=1

∑σ∈E(i)

S

⎡⎣ ∑ε=Dσ|Dσ′

ν hd−2ε

(un+1


)⎤⎦ un+1σ,i =

d∑i=1

∑ε=Dσ|Dσ′∈E(i)

S

ν hd−2ε

(un+1


)2

.

One may wonder why we do not use in Sn+1K the expression of this term as it is written in the remainder

Rn+1σ,i , i.e., in other words, use the numerical diffusion multiplied by u instead of the dissipation. A first answer

is that we mimic what happens at the continuous level: the term which appears in the kinetic energy balanceis div

(τ (u)

)· u and the corresponding term in the internal energy balance is the dissipation τ (u) : ∇u. A

more involved argument is that the expression in Sn+1K provides a positive source term to the internal energy

balance, and we may hope that the difference between the numerical diffusion multiplied by u and the associateddissipation tends to zero (because the numerical diffusion tends to zero) in the sense of distributions. To havean intuition of this fact, let us consider the toy elliptic problem, posed over Ω:

v − νΔv = f,

where ν is a positive parameter and f ∈ L2(Ω). Assuming homogeneous Dirichlet boundary conditions, weobtain by standard variational arguments ‖v‖L2(Ω) + ν1/2‖∇v‖L2(Ω)d ≤ C, with C only depending on Ω and f .We thus get, with ϕ ∈ C∞

c (Ω):∫Ω

[ν(Δv)v + ν|∇v|2

]ϕdx = ν

∫Ω

div(v∇v)ϕdx = −ν∫

Ω

v∇v · ∇ϕdx,


and so, finally, by the Cauchy−Schwarz inequality:∣∣∣∣∫Ω

[ν(Δv)v + ν|∇v|2

]ϕdx

∣∣∣∣ ≤ C ‖∇ϕ‖L∞(Ω)d ν1/2,

so this term tends to zero if so does ν. A discrete analogue of this simple computation is used to pass to thelimit in the scheme in the next section (with a control on the unknown assumed and not proven).

Note that the term Sn+1K is non-negative. Consequently, adapting the proof of ([18], Thm. 4.1) to cope with

this additional term, we obtain that the scheme admits at least one solution, which satisfies p ≥ 0, ρ ≥ 0 ande ≥ 0. In addition, equation (4.7) shows that the scheme conserves the integral of the total energy over thecomputational domain.

Theorem 4.1 (Existence and stability). Assume that for all K ∈ M, e0K > 0 and ρ−1K > 0. Then there exists

a solution to the scheme (4.3), which furthermore satisfies ρ0K > 0 and, for 1 ≤ n ≤ N and K ∈ M, en

K > 0,ρn

K > 0, and the following discrete analogue of the total energy balance:

∑K∈M

|K| enK +

12

d∑i=1

∑σ∈E(i)

S

|Dσ| ρn−1Dσ

(unσ,i)

2 + Rn

≤∑

K∈M|K| e0K +

12

d∑i=1

∑σ∈E(i)

S

|Dσ| ρ−1Dσ

(u0σ,i)

2 + R0,

where:

Rn = δt2∑

σ∈Eint

|Dσ|ρn−1

Dσ

|(∇p)nσ |2 = δt2

∑σ=K|L∈Eint

|σ|2

|Dσ| ρn−1Dσ

(pnK − pn

L)2.

4.3. Passing to the limit in the scheme (1D case)

We now undertake the passage to the limit in the scheme (4.3) in the one-dimensional case. Using thenotations (3.25)–(3.27), the scheme may be rewritten in “one-dimensional notations” as follows:

Initialization – Compute ρ−1, u0 and ρ0 by (3.53a) and e0 and p0 by:

∀K ∈ M, e0K =1|K|

∫K

e0(x) dx,

∀K ∈ M, p0K = (γ − 1) ρ0

K e0K .

(4.8a)

Pressure gradient renormalization step

∀σ = K|L ∈ Eint, (δp)n+1σ =

√ρn

Dσ

ρn−1Dσ

(pnK − pn

L) . (4.8b)


∀σ =−−→K|L ∈ Eint,

|Dσ|δt

(ρnDσun+1

σ − ρn−1Dσ

unσ) + Fn

L un+1L − Fn

K un+1K − |Dσ| (ΔMu)n+1

σ + (δp)n+1σ = 0. (4.8c)


Correction step – Solve for ρn+1, pn+1, en+1 and un+1:

∀σ =−−→K|L ∈ Eint,

|Dσ|δt

ρnDσ

(un+1σ − un+1

σ ) + (pn+1L − pn+1

K ) − (δp)n+1σ = 0, (4.8d)

∀K = [−→σσ′] ∈ M,

|K|δt

(ρn+1K − ρn

K) + Fn+1σ′ − Fn+1

σ = 0, (4.8e)

∀K = [−→σσ′] ∈ M,

|K|δt

(ρn+1K en+1

K − ρnKe

nK) + Fn+1

σ′ en+1σ′ − Fn+1

σ en+1σ

+pn+1K (un+1

σ′ − un+1σ ) = Sn+1

K ,(4.8f)

∀K ∈ M, pn+1K = (γ − 1) ρn+1

K en+1K . (4.8g)

The corrective term Sn+1K reads:

∀K = [σσ′], Sn+1K =

|K|4 δt

ρn−1K

[(un+1

σ − unσ

)2+(un+1

σ′ − unσ′)2]

+ν

hK

(un+1

σ − un+1σ′)2. (4.9)

For discrete functions q and v defined on the primal and dual mesh, respectively, we define a discreteL1((0, T ); BV (Ω)) norm by:

‖q‖T ,x,BV =N∑

n=0

δt∑

σ=K|L∈Eint

|qnL − qn

K |, ‖v‖T ,x,BV =N∑

n=0

δt∑

ε=Dσ|Dσ′∈Eint

|vnσ′ − vn

σ |,

and a discrete L1(Ω; BV ((0, T ))) norm by:

‖q‖T ,t,BV =∑

K∈M|K|

N−1∑n=0

|qn+1K − qn

K |, ‖v‖T ,t,BV =∑σ∈E

|Dσ|N−1∑n=0

|vn+1σ − vn

σ |.

For the proof of the consistency of the scheme (i.e. the proof of the Theorem 4.3 below), we need to introducethe following stability assumptions on a sequence (ρ(m), p(m), e(m), u(m), u(m))m∈N of discrete solutions:

|(ρ(m))nK | + |(p(m))n

K | + |(e(m))nK | ≤ C, ∀K ∈ M(m), for 0 ≤ n ≤ N (m), ∀m ∈ N, (4.10a)

1|(ρ(m))n

K | ≤ C, ∀K ∈ M(m), for 0 ≤ n ≤ N (m), ∀m ∈ N, (4.10b)

|(u(m))nσ| + |(u(m))n

σ| ≤ C, ∀σ ∈ E(m), for 0 ≤ n ≤ N (m), ∀m ∈ N, (4.10c)

‖ρ(m)‖T ,x,BV + ‖e(m)‖T ,x,BV + ‖p(m)‖T ,x,BV + ‖u(m)‖T ,x,BV ≤ C, ∀m ∈ N, (4.10d)

‖ρ(m)‖T ,t,BV + ‖u(m)‖T ,t,BV ≤ C, ∀m ∈ N. (4.10e)

Note that we do not suppose any control on time discrete derivatives of u(m) (i.e. on ‖u(m)‖T ,t,BV ) and on thediscrete space derivatives of u(m) (i.e. on ‖u(m)‖T ,x,BV ). Note also that, by definition of the initial conditionsof the scheme, these inequalities imply that the functions ρ0, e0 and u0 belong to L∞(Ω) ∩ BV (Ω). As in thebarotropic case, we are not able to prove (4.10) for the solutions of the scheme; however, such inequalities aresatisfied by the “interpolation” (for instance, by taking the cell average) of the solution to a Riemann problem,and are also observed in computations.

The following definition gathers the assumptions on the discretization which are needed in the proof ofTheorem 4.3 below.


Definition 4.2 (Regular sequence of discretizations). We define a regular sequence of discretizations(M(m), δt(m), ν(m))m∈N as a sequence of meshes, time steps and numerical diffusion coefficients which areassumed to satisfy the following conditions:

(i) both the time step δt(m) and the size h(m) of the mesh M(m) tend to zero as m→ +∞;(ii) there exists C ∈ R+ (not necessarily lower than 1) such that the following CFL-like condition holds:

∀m ∈ N,δt(m)


σ=K|L∈E(m)int

12

(hK + hL); (4.11)

(iii) there exists C ∈ R+ such that the sequence of numerical diffusion coefficients (ν(m))m∈N satisfies

limm→+∞

ν(m)

h(m)≤ C.

A weak solution to the continuous problem satisfies, for any ϕ ∈ C∞c

([0, T )×Ω

):

−∫

Ω×(0,T )

[ρ ∂tϕ+ ρ u ∂xϕ

]dxdt−

∫Ω

ρ0(x)ϕ(x, 0) dx = 0, (4.12a)

−∫

Ω×(0,T )

[ρ u ∂tϕ+ (ρ u2 + p) ∂xϕ

]dxdt−

∫Ω

ρ0(x)u0(x)ϕ(x, 0) dx = 0, (4.12b)

−∫

Ω×(0,T )

[ρE ∂tϕ+ (ρE + p)u ∂xϕ

]dxdt−

∫Ω

ρ0(x)E0(x)ϕ(x, 0) dx = 0, (4.12c)

p = (γ − 1)ρ e, E =12u2 + e, E0 =

12u2

0 + e0. (4.12d)

As in the barotropic case, these relations are not sufficient to define a weak solution to the problem, sincethey do not imply anything about the boundary conditions, but they allow to derive the Rankine–Hugoniotconditions.

We are now in position to state the following consistency result.

Theorem 4.3. Let Ω be an open bounded interval of R. Let (M(m), δt(m), ν(m))m∈N be a regular sequence ofdiscretizations in the sense of Definition 4.2. Let (ρ(m), p(m), e(m), u(m), u(m))m∈N be the corresponding sequenceof solutions. We suppose that this sequence satisfies the bounds (4.10) and converges in Lp(Ω × (0, T ))5, for1 ≤ p < +∞, to (ρ, p, e, ¯u, u) ∈ L∞(Ω × (0, T ))5.

Then ¯u = u and (ρ, p, e, u) satisfies the system (4.12).

Proof. Let us first check that ¯u = u. We first note that thanks to assumptions (4.10a) and (4.10b), the pressureprediction step (4.8b) yields that|(δp)n+1

σ | ≤ C |pnK − pn

L|. Therefore, from the expression of the correction stepfor the velocity (4.8d), we have, again using assumption (4.10b):

‖u(m) − u(m)‖L1(Ω×(0,T )) ≤ Cδt ‖p(m)‖T ,x,BV

which, passing to the limit when m→ +∞, yields the result.We now observe that the stability assumptions (4.10) and the regularity assumptions for the discretization

of Definition 4.2 are stronger than the hypotheses made in the barotropic case (see Def. 3.5, Rem. 3.19 andthe assumptions of Thm. 3.20). In particular, combining L∞, space BV estimates and the assumption (iii) ofDefinition 4.2 on the numerical diffusion coefficient, we can prove the same control on the space translates ofρ and u as provided by the remainder terms (3.48) of the inequality (3.47). Consequently, the passage to thelimit in the scheme for the mass and momentum balance equations is the same as in the barotropic case.


We thus only need to prove that (ρ, p, e, u) satisfies (4.12c). Let us first multiply the one dimensional versionof the discrete kinetic energy equation (3.44) by δt ϕn

σ and sum over the faces and the time steps. Similarly, wemultiply the discrete internal energy equation (4.8f) by δt ϕn

K , and sum over the primal cells and the time steps.Summing the two obtained relations, we get T (m)

1 + T(m)2 +T

(m)3 + T

(m)4 +T

(m)5 = R(m), where the terms T (m)

1 ,T

(m)3 , and T (m)

5 are defined by (3.37a)–(3.37c) in the proof of Theorem 3.20, and

T(m)2 =

N−1∑n=0

δt∑

K∈M

|K|δt

[ρn+1

K en+1K − ρn

KenK

]ϕn

K , (4.13)

T(m)4 =

N−1∑n=0

δt∑

K=[−−→σσ′]∈M

[Fn+1

σ′ en+1σ′ − Fn+1

σ en+1σ

]ϕn

K , (4.14)

R(m) = −N−1∑n=0

δt∑σ∈E

(Rn+1

σ − Pn+1σ

)ϕn

σ +N−1∑n=0

δt∑

K∈MSn+1

K ϕnK (4.15)

where, in the latter relation, the remainder term reads, for σ =−−→K|L ∈ Eint with K = [

−→σ′σ] and L = [

−−→σσ′′]:

Rn+1σ =

12|Dσ|δt

ρn−1Dσ

[un+1

σ − unσ

]2+ un+1

σ

[ν

hK(un+1

σ − un+1σ′ ) +

ν

hL

(un+1

σ − un+1σ′′)],

and where Pn+1σ is defined by (3.45), and Sn+1

K by (4.9).The study of the terms T (m)

1 , T (m)3 and T (m)

5 has already been performed in the proof of Theorem 3.20. Thestudy of the term T

(m)2 (resp. T (m)

4 ) is similar to that of the term T(m)2 (resp. T (m)

4 ) of the proof of Theorem 3.20,replacing the elastic potential H(ρ) by the internal energy e. Finally, we have to deal with the term R(m). Here,the situation is different from Theorem 3.20 since we have to prove that this term tends to zero whereas for theentropy inequality in the barotropic case, we only had to prove that it is non-negative at the limit of vanishingtime and space steps. We split R(m) into three terms: R(m) = R

(m)δt + R

(m)Δ + P (m) and give the expression of

each of these three terms hereafter. The first term reads:

R(m)δt = −1

2

N−1∑n=0

∑σ∈E

|Dσ| ρn−1Dσ

(un+1σ − un

σ)2 ϕnσ +

12

N−1∑n=0

∑K∈M

∑σ∈E(K)

|DK,σ| ρn−1K (un+1

σ − unσ)2 ϕn

K .

Thanks to the definition of the density on the faces, we get:

R(m)δt =

12

N−1∑n=0

∑K∈M

∑σ∈E(K)


σ − unσ)2 (ϕn

K − ϕnσ),

and therefore, thanks to the regularity of ϕ:

∣∣∣R(m)δt

∣∣∣ ≤ Cϕ hN−1∑n=0

∑K∈M

∑σ∈E(K)


σ − unσ)2.

Using the assumed uniform bound in L∞(Ω× (0, T )) for the sequence (ρ(m))m∈N, we obtain that the remainderR

(m)δt satisfies |R(m)

δt | ≤ C h(m)(R

(m)δt,1 +R

(m)δt,2

), where C is a real positive number, independent of m and

R(m)δt,1 =

N−1∑n=0

∑σ∈Eint

|Dσ| (un+1σ − un

σ)2, R(m)δt,2 =

N−1∑n=0

∑σ∈Eint

|Dσ|(un+1

σ − un+1σ

)2.


The first of this term satisfies ∣∣∣R(m)δt,1

∣∣∣ ≤ 2 ‖u(m)‖L∞(Ω×(0,T )) ‖u(m)‖T ,t,BV ,

and, thanks once again to the expression of the velocity correction step (4.8d), we have for the second one:∣∣∣R(m)δt,2

∣∣∣ ≤ C ‖ 1ρ(m)

‖L∞(Ω×(0,T ))

(‖u(m)‖L∞(Ω×(0,T )) + ‖u(m)‖L∞(Ω×(0,T ))

)‖p(m)‖T ,x,BV ,

where C is again a real positive number, independent of m. Hence R(m)δt tends to zero as m tends to +∞. Let

us now turn to R(m)Δ , which reads:

R(m)Δ = −

N−1∑n=0

δt∑


K=[−−→σ′σ], L=[

−−→σσ′′]

un+1σ

[ν

hK(un+1

σ − un+1σ′ ) +

ν

hL(un+1

σ − un+1σ′′ )]ϕn

σ

+N−1∑n=0

δt∑

K=[σσ′]∈M

ν

hK

(un+1

σ − un+1σ′)2

ϕnK .

As explained in Section 4.2, the general idea is now to recast this term as a discrete version of the integral overspace and time of a quantity of the form −u ∂xu ∂xϕ scaled by a numerical viscosity vanishing with the spacestep; then, the supposed controls on the solution imply that the term tends to zero. Reordering the sums, weget:

R(m)Δ =

N−1∑n=0

δt∑

K=[−−→σσ′]∈M

ν

hK(un+1

σ − un+1σ′ ) (un+1

σ′ ϕnσ′ − un+1

σ ϕnσ) +

ν

hK

(un+1

σ′ − un+1σ

)2ϕn

K ,

and thus:

R(m)Δ =

N−1∑n=0

δt∑

K=[−−→σσ′]∈M

ν

hK(un+1

σ − un+1σ′ )

[un+1

σ (ϕnK − ϕn

σ) + un+1σ′ (ϕn

σ′ − ϕnK)].

We thus get, thanks to the regularity of ϕ:

|R(m)Δ | ≤ ν Cϕ ‖u(m)‖L∞(Ω×(0,T )) ‖u(m)‖T ,x,BV ,

which yields the desired estimate. Finally, P (m) reads:

P (m) =N−1∑n=0

δt∑σ∈E

Pn+1σ ϕn

σ,

and the fact that this term tends to zero is stated in Lemma 3.16. This concludes the proof. �

4.4. Numerical tests

In this section, we assess the behaviour of the scheme on a one dimensional Riemann problem. We chooseinitial conditions such that the structure of the solution consists in two shock waves, separated by the contactdiscontinuity, with sufficiently strong shocks to allow an easy discrimination of correct numerical solutions.These initial conditions are those proposed in ([63], Chap. 4), for the test referred to as Test 5:

left state:

⎡⎣ρleft

uleft

pleft

⎤⎦ =

⎡⎣ 5.9992419.5975460.894

⎤⎦ right state:

⎡⎣ρright

uright

pright

⎤⎦ =

⎡⎣ 5.99242−6.1963346.0950

⎤⎦.The problem is posed over Ω = (−0.5, 0.5), and the discontinuity is initially located at x = 0.


At the boundaries, since, in this test, the flow is entering the domain, the solution is prescribed (which,in fact, is unimportant, the solution being constant at any time in a sufficiently large neighborhood of theseboundaries). Previous numerical experiments addressing barotropic flows [27] showed that, at least for onedimensional computations with schemes similar to the one under study here, it was not necessary to upwind theconvection term in the momentum balance equation; consequently, we only employ a centered approximationof the velocity at the dual faces.

The computations are performed with the open-source software CALIF3S [5].The density fields obtained with h = 1/2000 (or a number of cells n = 2000) at t = 0.035, with and without

assembling the corrective source term in the internal energy balance (SK)K∈M, together with the analyticalsolution, are shown on Figure 2. The density and the pressure obtained, still with and without corrective terms,for various meshes, are plotted on Figures 3 and 4 respectively. For these computations, we take δt = h/20,which yields a cfl number, with respect to the material velocity only, close to one. The first conclusion is thatboth schemes seem to converge, but the corrective term is necessary to obtain the right solution. In this case, forinstance, we obtain the correct intermediate state for the pressure and velocity up to four digits in the essentialpart of the corresponding zone:

(analytical) intermediate state:[p∗

u∗

]=[1691.658.68977

]for x ∈ (0.028, 0.428)

numerical results:

∣∣∣∣∣p ∈ (1691.6, 1691.8)

u ∈ (8.689, 8.690)for x ∈ (0.032, 0.417).

Without a corrective term, one can check that the obtained solution is not a weak solution to the Eulersystem: indeed, the Rankine–Hugoniot condition applied to the total energy balance, with the states obtainednumerically, yields a right shock speed slightly greater than the analytical solution one, while the same shockspeed obtained numerically is clearly lower.

We also observe that the scheme is rather diffusive especially for contact discontinuities for which the beneficialcompressive effect of the shocks does not apply. More accurate variants may certainly be derived, using forinstance MUSCL-like techniques; this work is underway [17].

In order to check the consistency of the scheme, we give in the table below the L1(Ω)-norm of the differencebetween the numerical and the exact solution (denoted (ρex, pex, uex)) at t = 0.035, for various grids and stillfor δt = h/20.

h ‖ρ− ρex‖L1(Ω) ‖p− pex‖L1(Ω) ‖u− uex‖L1(Ω)

1/250 0.0662 1.235 0.009111/500 0.0452 0.619 0.004371/1000 0.0313 0.365 0.002321/2000 0.0215 0.170 0.001251/4000 0.0148 0.0849 0.0006251/8000 0.0102 0.0357 0.000358

We observe a convergence rate of approximatively 1 for the variables which are continuous at the contactdiscontinuity (p and u) and 1/2 for the other ones (in the table, only ρ). Since, as explained above, the erroressentially lies at the jumps of the solution, it means that a shock is captured in approximatively the samenumber of cells, for any of the meshes used in this test; by a simple computation, this implies that the correctiveterm (SK)K∈M does not tend to zero.

5. Conclusion, perspectives

In this work, we studied staggered implicit and semi-implicit schemes which had been found earlier to bevery efficient for viscous compressible flows [14–16]. We were able to show here that they are also well adapted


0

5

10

15

20

25

30

35

40

-0.4 -0.2 0 0.2 0.4

dens

ity

x

withwithout

exact

Figure 2. Test 5 of [63], Chapter 4 - Density obtained with n = 2000 cells, with and withoutcorrective source terms, and analytical solution.

0

5

10

15

20

25

30

35

40

-0.4 -0.2 0 0.2 0.4

dens

ity

x

n=500n=1000n=2000

0

5

10

15

20

25

30

35

40

-0.4 -0.2 0 0.2 0.4

dens

ity

x

n=500n=1000n=2000

Figure 3. Test 5 of ([63], Chap. 4) – Density obtained with various meshes, with (left) andwithout (right) corrective source terms.

to the shallow water equations or barotropic Euler equations. We also showed that, with a careful discretizationof the internal energy equation, they are again an efficient choice for the full Euler equations.

There are several open questions under study at the present time or that will be in the near future:

– A proof of the consistency of the scheme in the multidimensional case is underway: there is a real difficulty ingoing from the 1D case that we studied here in Sections 3.1.3, 3.2.3 and 4.3 to the multidimensional case, dueto the intricate definition of the nonlinear convection term in the momentum balance equation, which makescomplex the passage to the limit in this term. This difficulty has been adressed both for the fluxes defined onunstructured meshes, with the Crouzeix−Raviart and Rannacher−Turek finite element unknowns [42], andin the MAC case [29].


0

200

400

600

800

1000

1200

1400

1600

1800

-0.4 -0.2 0 0.2 0.4

pres

sure

x

n=500n=1000n=2000

0

200

400

600

800

1000

1200

1400

1600

1800

-0.4 -0.2 0 0.2 0.4

pres

sure

x

n=500n=1000n=2000

Figure 4. Test 5 of ([63], Chap. 4) – pressure obtained with various meshes, with (left) andwithout (right) corrective source terms.

– The consistency of the scheme with the entropy condition in the case of the full Euler equation is an open ques-tion, although the numerical results suggest that this is true. We shall continue to investigate this importantissue.

– Comparisons are currently being performed, for the proposed pressure correction scheme, between the presentstaggered discretization and a colocated version (density, pressure and velocity unknowns all in the cells), forthe Euler equations [62].

– An explicit version of this scheme has been studied both for the shallow water and Euler equations [30].For this time discretization, higher order schemes using a MUSCL technique [56] have been studied andimplemented [17].

– Finally, setting ρ to a constant in the pressure correction scheme yields the usual projection scheme forthe incompressible Euler (or Navier–Stokes) equations. A natural question is therefore to know whether thescheme is indeed asymptotic preserving: does the approximate solution tend to the incompressible solutionas the Mach number tends to 0? This question should be addressed in the near future.

Appendix A. Some results associated

with finite volume convection operators

We gather in this section some results concerning the finite volume discretization of the two convectionoperators which appear in the Navier–Stokes equations:

– the convection operator C appearing in the mass balance, which reads, at the continuous level,

ρ �→ C(ρ) = ∂tρ+ div(ρu), (A.1)

where u stands for a given velocity field (which is not assumed to satisfy any divergence constraint),– the convection operator Cρ appearing in the momentum and energy balances, which reads, in the contin-

uous setting,z �→ Cρ(z) = ∂t(ρz) + div(ρzu), (A.2)

where ρ (resp. u) stands for a given scalar (resp. vector) field; we wish to obtain some property of Cρ

under the assumption that ρ and u satisfy a mass balance equation, i.e. ∂tρ+ div(ρu) = 0.Multiplying these operators by functions depending on the unknown is a classical technique to obtain convectionoperators acting over different variables, possibly with residual terms: one may think, for instance, to the theory


of renormalized solutions or entropy solutions for the operator C, or, in fluid mechanics, to the derivation of theso-called kinetic energy transport identity for the operator Cρ, with z standing for a component of the velocity.The results provided in this section are the discrete analogs of these properties.

We begin with a property of the convection operator C defined by (A.1); at the continuous level, this propertymay be formally obtained as follows. Let ψ be a regular function from (0,+∞) to R; then:

ψ′(ρ) C(ρ) = ψ′(ρ) ∂t(ρ) + ψ′(ρ)u · ∇ρ+ ψ′(ρ) ρ divu = ∂t(ψ(ρ)) + u · ∇ψ(ρ) + ρψ′(ρ) divu;

adding and subtracting ψ(ρ) divu yields:

ψ′(ρ) C(ρ) = ∂t

(ψ(ρ))

+ div(ψ(ρ)u

)+(ρψ′(ρ) − ψ(ρ)

)divu. (A.3)

This computation is of course completely formal and only valid for regular functions ρ and u. The followinglemma states a discrete analogue to (A.3), and its proof follows the formal computation which we just described.

Lemma A.1. Let P be a polygonal (resp. polyhedral) bounded set of R2 (resp. R3), and let E(P ) be the set ofits edges (resp. faces). Let ψ be a continuously differentiable function defined over (0,+∞). Let ρ∗P > 0, ρP > 0,δt > 0; consider three families (ρη)η∈E(P ) ⊂ R+ \ {0}, (Vη)η∈E(P ) ⊂ R and (Fη)η∈E(P ) ⊂ R such that

∀η ∈ E(P ), Fη = ρη Vη,

and define:

RP,δt =

⎡⎣ |P |δt

(ρP − ρ∗P ) +∑

η∈E(P )

Fη

⎤⎦ψ′(ρP ) (A.4)

−

⎡⎣ |P |δt

(ψ(ρP ) − ψ(ρ∗P )

)+∑

η∈E(P )

ψ(ρη) Vη + (ρPψ′(ρP ) − ψ(ρP ))

∑η∈E(P )

Vη

⎤⎦ (A.5)

Then

(i) If ψ is convex and ρη = ρP whenever Vη > 0, then RP,δt ≥ 0.(ii) If ψ is twice continuously differentiable then

RP,δt =12|P |δt

ψ′′(ρP ) (ρP − ρ∗P )2 − 12

∑η∈E(P )

Vη ψ′′(ρη) (ρη − ρP )2,

with ρP ∈ [min(ρP , ρ∗P ),max(ρP , ρ

∗P )] and ∀η ∈ E(P ), ρη ∈ [min(ρη, ρP ),max(ρη, ρP )].

Proof. We have:⎡⎣ |P |δt

(ρP − ρ∗P ) +∑

η∈E(P )

Fη

⎤⎦ ψ′(ρP ) =|P |δt

(ρP − ρ∗P ) ψ′(ρP ) +∑

η∈E(P )

ψ(ρη) Vη

+∑

η∈E(P )

[ρηψ′(ρP ) − ψ(ρη)]Vη,

so the remainder term RP,δt reads RP,δt =|P |δt

rP +∑

η∈E(P ) Vη rη, with:

rP = (ρP − ρ∗P ) ψ′(ρP ) −[ψ(ρP ) − ψ(ρ∗P )

], rη = ρηψ

′(ρP ) − ψ(ρη) −[ρPψ

′(ρP ) − ψ(ρP )].


If the function ψ is convex, rP is non-negative while rη is non-positive (and vanishes if ρη = ρP ). If ψ is twicecontinuously differentiable, a Taylor expansion gives that:

(ρP − ρ∗P )ψ′(ρP ) = ψ(ρP ) − ψ(ρ∗P ) +12ψ

′′(ρP )(ρP − ρ∗P )2,

ρηψ′(ρP ) − ψ(ρη) = ρPψ

′(ρP ) − ψ(ρP ) − 12ψ′′(ρη)(ρη − ρP )2,

with ρP ∈ [min(ρP , ρ∗P ),max(ρP , ρ

∗P )] and for any η ∈ E(P ), ρη ∈ [min(ρη, ρP ),max(ρη, ρP )]; hence the re-

sult. �

We now turn to the momentum convection operator Cρ defined by (A.2); formally, using twice the assumption∂tρ+ div(ρu) = 0 yields:

ψ′(z) Cρ(z) =ψ′(z)[∂t(ρ z) + div(ρ z u)

]= ψ′(z)ρ

[∂tz + u · ∇z

]= ρ[∂tψ(z) + u · ∇ψ(z)

]= ∂t

(ρψ(z)

)+ div

(ρψ(z)u

).

Taking for z a component of the velocity field, this relation is the central argument used to derive the kineticenergy balance. The following lemma states a discrete counterpart of this identity.

Lemma A.2. Let P be a polygonal (resp. polyhedral) bounded set of R2 (resp. R3) and let E(P ) be the set ofits edges (resp. faces). Let ρ∗P > 0, ρP > 0, δt > 0, and (Fη)η∈E(P ) ⊂ R be such that

|P |δt

(ρP − ρ∗P ) +∑

η∈E(P )

Fη = 0. (A.6)

Let ψ be a continuously differentiable function defined over (0,+∞). For u∗P ∈ R, uP ∈ R and (uη)η∈E(P ) ⊂ R

let us define:

RP,δt =

⎡⎣ |P |δt

(ρP uP − ρ∗P u

∗P

)+∑

η∈E(P )

Fη uη

⎤⎦ ψ′(uP )

−

⎡⎣ |P |δt

[ρP ψ(uP ) − ρ∗P ψ(u∗P )

]+∑

η∈E(P )

Fη ψ(uη)

⎤⎦ . (A.7)

Then:

(i) If ψ is convex and uη = uP whenever Fη > 0, then RP,δt ≥ 0.(ii) If ψ is twice continuously differentiable, then

RP,δt =12|P |δtρ∗P ψ′′(uP )(uP − u∗P )2 − 1

2

∑η∈E(P )

Fη ψ′′(uη) (uη − uP )2, (A.8)

with, uP ∈ [min(uP , u∗P ),max(uP , u

∗P )] and, ∀η ∈ E(P ), uη ∈ [min(uη, uP ),max(uη, uP )].

(iii) As a consequence of (ii), for ψ defined by ψ(s) = s2/2, and ∀η ∈ E(P ), uη such that uη = (uP + uPη )/2(this is simply obtained by defining uPη = 2 uη − uP ), we get the following identity:⎡⎣ |P |

δt(ρP uP − ρ∗P u

∗P ) +

∑η∈E(P )

Fη uη

⎤⎦ uP =12

|P |δt

[ρP u

2P − ρ∗P (u∗P )2

]+

12

∑η∈E(P )

Fη uP uPη + RP,δt,

(A.9)with

RP,δt =12|P |δtρ∗P (uP − u∗P )2,


Proof. Let TP be defined by:

TP =

⎡⎣ |P |δt

(ρPuP − ρ∗Pu∗P ) +

∑η∈E(P )

Fη uη

⎤⎦ψ′(uP ).

Using equation (A.6), we obtain:

TP =

⎡⎣ |P |δt

ρ∗P (uP − u∗P ) +∑

η∈E(P )

Fη(uη − uP )

⎤⎦ψ′(uP ).

We now define the remainder terms rP and (rη)η∈E(P ) by:

rP = (uP − u∗P ) ψ′(uP ) −[ψ(uP ) − ψ(u∗P )

], rη = (uP − uη) ψ′(uP ) −

[ψ(uP ) − ψ(uη)

].

With these notations, we get:

TP =|P |δt

ρ∗P[ψ(uP ) − ψ(u∗P )

]+∑

η∈E(P )

Fη

[ψ(uη) − ψ(uP )

]+

|P |δt

ρ∗P rP −∑

η∈E(P )

Fη rη.

Using equation (A.6) once again, we have:

TP =|P |δt

[ρP ψ(uP ) − ρ∗P ψ(u∗P )

]+∑

η∈E(P )

Fη ψ(uη) +|P |δt

ρ∗P rP −∑

η∈E(P )

Fη rη,

and thus:

RP,δt =|P |δt

ρ∗P rP −∑

η∈E(P )

Fη rη.

If ψ is convex, the remainder terms rP and (rη)η∈E(P ) are non-negative, and if uη = uP , rη = 0; hence, if wesuppose that uη = uP when Fη ≥ 0, then RP,δt ≥ 0. If ψ is twice continuously differentiable, a Taylor expansionyields:

rP =12ψ′′(uP ) (up − u∗p)

2, rη =12ψ′′(uη) (uη − up)2

with uP ∈ [min(uP , u∗P ),max(uP , u

∗P )] and, ∀η ∈ E(P ), uη ∈ [min(uη, uP ),max(uη, uP )]. Thus (ii) holds. The

assertion (iii) is then a direct consequence of (ii). �

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